1.
Trefoil knot fold
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The trefoil knot fold is a protein fold in which the protein backbone is twisted into a trefoil knot shape. In many cases the trefoil knot is part of the site or a ligand-binding site and is critical to the activity of the enzyme in which it appears. Before the discovery of the first knotted protein, it was believed that the process of protein folding could not efficiently produce deep knots in protein backbones. Studies of the kinetics of a dimeric protein from Haemophilus influenzae have revealed that the folding of trefoil knot proteins may depend on proline isomerization. Currently, there is a web server available to detect knots in proteins as well as to provide information on knotted proteins in the Protein Data Bank. Knottins are small, diverse and stable proteins with important drug design potential and they can be classified in 30 families which cover a wide range of sequences, three-dimensional structures and functions. Inter knottin similarity lies mainly between 20% and 40% sequence identity and 1.5 to 4 A backbone deviations although they all share a tightly knotted disulfide core and this important variability is likely to arise from the highly diverse loops which connect the successive knotted cysteines. Deep trefoil knot implicated in RNA binding found in an archaebacterial protein. ^ Nureki O, Shirouzu M, Hashimoto K, Ishitani R, Terada T, Tamakoshi M, Oshima T, Chijimatsu M, Takio K, Vassylyev DG, Shibata T, Inoue Y, Kuramitsu S, an enzyme with a deep trefoil knot for the active-site architecture. ^ Nureki O, Watanabe K, Fukai S, Ishii R, Endo Y, Hori H, deep knot structure for construction of active site and cofactor binding site of tRNA modification enzyme. ^ Leulliot N, Bohnsack MT, Graille M, Tollervey D, the yeast ribosome synthesis factor Emg1 is a novel member of the superfamily of alpha/beta knot fold methyltransferases. ^ Tkaczuk KL, Dunin-Horkawicz S, Purta E, Bujnicki JM, structural and evolutionary bioinformatics of the SPOUT superfamily of methyltransferases. Probing natures knots, the pathway of a knotted homodimeric protein. ^ Khatib F, Weirauch MT, Rohl CA, rapid knot detection and application to protein structure prediction. ^ Lai YL, Yen SC, Yu SH, Hwang JK. pKNOT, the protein KNOT web server
2.
Overhand knot
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The overhand knot is one of the most fundamental knots, and it forms the basis of many others, including the simple noose, overhand loop, anglers loop, reef knot, fishermans knot, and water knot. The overhand knot is a stopper, especially when used alone and it should be used if the knot is intended to be permanent. It is often used to prevent the end of a rope from unraveling, an overhand knot becomes a trefoil knot, a true knot in the mathematical sense, by joining the ends. There are a number of ways to tie the Overhand knot, thumb method – create a loop and push the working end through the loop with your thumb. Overhand method – create a bight, by twisting the hand over at the wrist and sticking your hand in the hole, pinch the working end with your fingers and pull through the loop. In heraldry, the knot is known as a Stafford knot, due to use first as a heraldic badge by the Lords of Stafford. As a defensive measure, hagfishes, which resemble eels, produce large volumes of thick slime when disturbed. A hagfish can remove the slime, which can suffocate it in a matter of minutes, by tying its own body into an overhand knot. This action scrapes the slime off the fishs body, hagfish also tie their bodies into overhand knots in order to create leverage to rip off chunks of their preys flesh, but do so in reverse. If the two ends of an overhand knot are joined together, this becomes equivalent to the trefoil knot of mathematical knot theory. If a flat ribbon or strip is folded into a flattened overhand knot. List of knots Trefoil knot, the treatment of the overhand knot Double overhand knot The Ashley Book of Knots
3.
Braid length
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In topology, a branch of mathematics, braid theory is an abstract geometric theory studying the everyday braid concept, and some generalizations. The idea is that braids can be organized into groups, in which the operation is do the first braid on a set of strings. Such groups may be described by explicit presentations, as was shown by Emil Artin, for an elementary treatment along these lines, see the article on braid groups. Braid groups are also understood by a deeper mathematical interpretation, as the group of certain configuration spaces. To explain how to reduce a braid group in the sense of Artin to a fundamental group and that is, an ordered n-tuple is in the same orbit as any other that is a re-ordered version of it. A path in the symmetric product is the abstract way of discussing n points of X, considered as an unordered n-tuple. Since we must require that the strings never pass through other, it is necessary that we pass to the subspace Y of the symmetric product. That is, we remove all the subspaces of Xn defined by conditions xi = xj and this is invariant under the symmetric group, and Y is the quotient by the symmetric group of the non-excluded n-tuples. Under the dimension condition Y will be connected, with this definition, then, we can call the braid group of X with n strings the fundamental group of Y. The case where X is the Euclidean plane is the one of Artin. In some cases it can be shown that the homotopy groups of Y are trivial. When X is the plane, the braid can be closed, i. e. corresponding ends can be connected in pairs, to form a link, i. e. a possibly intertwined union of possibly knotted loops in three dimensions. The number of components of the link can be anything from 1 to n, a theorem of J. W. Alexander demonstrates that every link can be obtained in this way as the closure of a braid. Different braids can give rise to the link, just as different crossing diagrams can give rise to the same knot. Markov describes two moves on braid diagrams that yield equivalence in the corresponding closed braids, a single-move version of Markovs theorem, was published by Lambropoulou & Rourke. Vaughan Jones originally defined his polynomial as an invariant and then showed that it depended only on the class of the closed braid. The braid index is the least number of strings needed to make a closed braid representation of a link and it is equal to the least number of Seifert circles in any projection of a knot. Additionally, the length is the longest dimension of a braid
4.
Braid number
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In topology, a branch of mathematics, braid theory is an abstract geometric theory studying the everyday braid concept, and some generalizations. The idea is that braids can be organized into groups, in which the operation is do the first braid on a set of strings. Such groups may be described by explicit presentations, as was shown by Emil Artin, for an elementary treatment along these lines, see the article on braid groups. Braid groups are also understood by a deeper mathematical interpretation, as the group of certain configuration spaces. To explain how to reduce a braid group in the sense of Artin to a fundamental group and that is, an ordered n-tuple is in the same orbit as any other that is a re-ordered version of it. A path in the symmetric product is the abstract way of discussing n points of X, considered as an unordered n-tuple. Since we must require that the strings never pass through other, it is necessary that we pass to the subspace Y of the symmetric product. That is, we remove all the subspaces of Xn defined by conditions xi = xj and this is invariant under the symmetric group, and Y is the quotient by the symmetric group of the non-excluded n-tuples. Under the dimension condition Y will be connected, with this definition, then, we can call the braid group of X with n strings the fundamental group of Y. The case where X is the Euclidean plane is the one of Artin. In some cases it can be shown that the homotopy groups of Y are trivial. When X is the plane, the braid can be closed, i. e. corresponding ends can be connected in pairs, to form a link, i. e. a possibly intertwined union of possibly knotted loops in three dimensions. The number of components of the link can be anything from 1 to n, a theorem of J. W. Alexander demonstrates that every link can be obtained in this way as the closure of a braid. Different braids can give rise to the link, just as different crossing diagrams can give rise to the same knot. Markov describes two moves on braid diagrams that yield equivalence in the corresponding closed braids, a single-move version of Markovs theorem, was published by Lambropoulou & Rourke. Vaughan Jones originally defined his polynomial as an invariant and then showed that it depended only on the class of the closed braid. The braid index is the least number of strings needed to make a closed braid representation of a link and it is equal to the least number of Seifert circles in any projection of a knot. Additionally, the length is the longest dimension of a braid
5.
Bridge number
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In the mathematical field of knot theory, the bridge number is an invariant of a knot defined as the minimal number of bridges required in all the possible bridge representations of a knot. Given a knot or link, draw a diagram of the using the convention that a gap in the line denotes an undercrossing. Call an arc in this diagram a bridge if it includes at least one overcrossing, then the bridge number of a knot can be found as the minimum number of bridges required for any diagram of the knot. Bridge number was first studied in the 1950s by Horst Schubert, the bridge number can equivalently be defined geometrically instead of topologically. In bridge representation, a knot lies entirely in the plane apart for a number of bridges whose projections onto the plane are straight lines. Equivalently the bridge number is the number of local maxima of the projection of the knot onto a vector. Every non-trivial knot has bridge number at least two, so the knots that minimize the number are the 2-bridge knots. It can be shown that every knot can be decomposed into two trivial n-tangles and hence 2-bridge knots are rational knots. If K is the sum of K1 and K2, then the bridge number of K is one less than the sum of the bridge numbers of K1. Crossing number Linking number Stick number Unknotting number Cromwell, Peter
6.
Crossing number (knot theory)
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In the mathematical area of knot theory, the crossing number of a knot is the smallest number of crossings of any diagram of the knot. By way of example, the unknot has crossing number zero, tables of prime knots are traditionally indexed by crossing number, with a subscript to indicate which particular knot out of those with this many crossings is meant. The listing goes 31,41,51,52,61 and this order has not changed significantly since P. G. Tait published a tabulation of knots in 1877. There has been little progress on understanding the behavior of crossing number under rudimentary operations on knots. A big open question if the crossing number is additive when taking knot sums. It is also expected that a satellite of a knot K should have larger crossing number than K, additivity of crossing number under knot sum has been proven for special cases, for example if the summands are alternating knots, or if the summands are torus knots. Marc Lackenby has also given a proof that there is a constant N >1 such that 1 N ≤ c r, but his method, there are connections between the crossing number of a knot and the physical behavior of DNA knots. For prime DNA knots, crossing number is a predictor of the relative velocity of the DNA knot in agarose gel electrophoresis. Basically, the higher the number, the faster the relative velocity. For composite knots, this not appear to be the case. There are related concepts of average crossing number and asymptotic crossing number, both of these quantities bound the standard crossing number. Asymptotic crossing number is conjectured to be equal to crossing number, other numerical knot invariants include the bridge number, linking number, stick number, and unknotting number
7.
Knot genus
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In mathematics, a Seifert surface is a surface whose boundary is a given knot or link. Such surfaces can be used to study the properties of the knot or link. For example, many knot invariants are most easily calculated using a Seifert surface, Seifert surfaces are also interesting in their own right, and the subject of considerable research. Specifically, let L be an oriented knot or link in Euclidean 3-space. Note that any compact, connected, oriented surface with nonempty boundary in Euclidean 3-space is the Seifert surface associated to its boundary link, a single knot or link can have many different inequivalent Seifert surfaces. A Seifert surface must be oriented and it is possible to associate surfaces to knots which are not oriented nor orientable, as well. The standard Möbius strip has the unknot for a boundary but is not considered to be a Seifert surface for the unknot because it is not orientable, the checkerboard coloring of the usual minimal crossing projection of the trefoil knot gives a Mobius strip with three half twists. As with the example, this is not a Seifert surface as it is not orientable. Applying Seiferts algorithm to this diagram, as expected, does produce a Seifert surface, in case, it is a punctured torus of genus g=1. It is a theorem that any link always has an associated Seifert surface and this theorem was first published by Frankl and Pontrjagin in 1930. A different proof was published in 1934 by Herbert Seifert and relies on what is now called the Seifert algorithm, the algorithm produces a Seifert surface S, given a projection of the knot or link in question. Suppose that link has m components, the diagram has d crossing points, then the surface S is constructed from f disjoint disks by attaching d bands. The homology group H1 is free abelian on 2g generators, the intersection form Q on H1 is skew-symmetric, and there is a basis of 2g cycles a1, a2. a2g with Q= the direct sum of g copies of. The 2g × 2g integer Seifert matrix V= has v the linking number in Euclidean 3-space of ai, every integer 2g × 2g matrix V with V − V * = Q arises as the Seifert matrix of a knot with genus g Seifert surface. The Alexander polynomial is computed from the Seifert matrix by A = d e t, the Alexander polynomial is independent of the choice of Seifert surface S, and is an invariant of the knot or link. The signature of a knot is the signature of the symmetric Seifert matrix V + V ⊤ and it is again an invariant of the knot or link. The genus of a knot K is the knot invariant defined by the genus g of a Seifert surface for K. For instance, An unknot—which is, by definition, the boundary of a genus zero
8.
Hyperbolic volume
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In the mathematical field of knot theory, the hyperbolic volume of a hyperbolic link is the volume of the links complement with respect to its complete hyperbolic metric. The volume is necessarily a finite number, and is a topological invariant of the link. As a link invariant, it was first studied by William Thurston in connection with his geometrization conjecture, the components of the link will become cusps of the 3-manifold, and the manifold itself will have finite volume. In particular, the volume of the complement is a knot invariant. In order to make it well-defined for all knots or links, there are only finitely many hyperbolic knots for any given volume. In practice, hyperbolic volume has proven effective in distinguishing knots. Jeffrey Weekss computer program SnapPea is the tool used to compute hyperbolic volume of a link. The Weeks manifold has the smallest possible volume of any closed manifold, Thurston and Jørgensen proved that the set of real numbers that are hyperbolic volumes of 3-manifolds is well-ordered, with order type ωω
9.
Stick number
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In the mathematical theory of knots, the stick number is a knot invariant that intuitively gives the smallest number of straight sticks stuck end to end needed to form a knot. Specifically, given any knot K, the number of K. Six is the lowest stick number for any nontrivial knot, there are few knots whose stick number can be determined exactly. Gyo Taek Jin determined the number of a -torus knot T in case the parameters p and q are not too far from each other, stick =2 q. The same result was found independently around the time by a research group around Colin Adams. Why knot, knots, molecules and stick numbers, Plus Magazine, an accessible introduction into the topic, also for readers with little mathematical background. The Knot Book, An elementary introduction to the theory of knots, Providence, RI, American Mathematical Society. Brennan, Bevin M. Greilsheimer, Deborah L. Woo, stick numbers and composition of knots and links, Journal of Knot Theory and its Ramifications,6, 149–161, doi,10. 1142/S0218216597000121, MR1452436. Calvo, Jorge Alberto, Geometric knot spaces and polygonal isotopy, Journal of Knot Theory and its Ramifications,10, 245–267, doi,10. 1142/S0218216501000834, MR1822491. Jin, Gyo Taek, Polygon indices and superbridge indices of torus knots and links, Journal of Knot Theory and its Ramifications,6, 281–289, doi,10. 1142/S0218216597000170, MR1452441. Negami, Seiya, Ramsey theorems for knots, links and spatial graphs, Transactions of the American Mathematical Society,324, 527–541, doi,10. 2307/2001731, MR1069741. Huh, Youngsik, Oh, Seungsang, An upper bound on stick number of knots, Journal of Knot Theory and its Ramifications,20, 741–747, doi,10. 1142/S0218216511008966, stick numbers for minimal stick knots, KnotPlot Research and Development Site
10.
Unknotting number
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In the mathematical area of knot theory, the unknotting number of a knot is the minimum number of times the knot must be passed through itself to untie it. If a knot has unknotting number n, then there exists a diagram of the knot which can be changed to unknot by switching n crossings, the unknotting number of a knot is always less than half of its crossing number. Any composite knot has unknotting number at least two, and therefore every knot with unknotting number one is a prime knot. The following table show the numbers for the first few knots, In general. Known cases include, The unknotting number of a nontrivial twist knot is equal to one. The unknotting number of a knot is equal to /2. The unknotting numbers of knots with nine or fewer crossings have all been determined. Crossing number Bridge number Linking number Stick number Unknotting problem Three_Dimensional_Invariants#Unknotting_Number, The Knot Atlas
11.
Conway notation (knot theory)
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In knot theory, Conway notation, invented by John Horton Conway, is a way of describing knots that makes many of their properties clear. It composes a knot using certain operations on tangles to construct it, in Conway notation, the tangles are generally algebraic 2-tangles. This means their tangle diagrams consist of 2 arcs and 4 points on the edge of the diagram, furthermore, tangles consisting only of positive crossings are denoted by the number of crossings, or if there are only negative crossings it is denoted by a negative number. If the arcs are not crossed, or can be transformed to into a position with the Reidemeister moves, it is called the 0 or ∞ tangle. If a tangle, a, is reflected on the NW-SE line, tangles have three binary operations, sum, product, and ramification, however all can be explained using tangle addition and negation. The tangle product, a b, is equivalent to −a+b. rational tangles are equivalent if and only if their fractions are equal. An accessible proof of this fact is given in, a number before an asterisk, *, denotes the polyhedron number, multiple asterisks indicate that multiple polyhedra of that number exist. Dowker notation Alexander–Briggs notation Conway, J. H, an Enumeration of Knots and Links, and Some of Their Algebraic Properties. In J. Leech, Computational Problems in Abstract Algebra, pdf available online Louis H. Kauffman, Sofia Lambropoulou, On the classification of rational tangles. Advances in Applied Mathematics,33, No
12.
Knot theory
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In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, in mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3. Knots can be described in various ways, given a method of description, however, there may be more than one description that represents the same knot. For example, a method of describing a knot is a planar diagram called a knot diagram. Any given knot can be drawn in different ways using a knot diagram. Therefore, a problem in knot theory is determining when two descriptions represent the same knot. A complete algorithmic solution to this exists, which has unknown complexity. In practice, knots are often distinguished by using a knot invariant, important invariants include knot polynomials, knot groups, and hyperbolic invariants. The original motivation for the founders of theory was to create a table of knots and links. Over six billion knots and links have been tabulated since the beginnings of knot theory in the 19th century, to gain further insight, mathematicians have generalized the knot concept in several ways. Knots can be considered in other spaces and objects other than circles can be used. Higher-dimensional knots are n-dimensional spheres in m-dimensional Euclidean space, archaeologists have discovered that knot tying dates back to prehistoric times. Besides their uses such as recording information and tying objects together, knots have interested humans for their aesthetics, Knots appear in various forms of Chinese artwork dating from several centuries BC. The endless knot appears in Tibetan Buddhism, while the Borromean rings have made repeated appearances in different cultures, the Celtic monks who created the Book of Kells lavished entire pages with intricate Celtic knotwork. Mathematical studies of knots began in the 19th century with Gauss, in the 1860s, Lord Kelvins theory that atoms were knots in the aether led to Peter Guthrie Taits creation of the first knot tables for complete classification. Tait, in 1885, published a table of knots with up to ten crossings and this record motivated the early knot theorists, but knot theory eventually became part of the emerging subject of topology. This would be the approach to knot theory until a series of breakthroughs transformed the subject. In the late 1970s, William Thurston introduced hyperbolic geometry into the study of knots with the hyperbolization theorem, many knots were shown to be hyperbolic knots, enabling the use of geometry in defining new, powerful knot invariants
13.
Dowker notation
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In the mathematical field of knot theory, the Dowker notation, also called the Dowker–Thistlethwaite notation or code, for a knot is a sequence of even integers. The notation is named after Clifford Hugh Dowker and Morwen Thistlethwaite, to generate the Dowker notation, traverse the knot using an arbitrary starting point and direction. Label each of the n crossings with the numbers 1, 2n in order of traversal, with the following modification, if the label is an even number and the strand followed crosses over at the crossing, then change the sign on the label to be a negative. When finished, each crossing will be labelled a pair of integers, one even, the Dowker notation is the sequence of even integer labels associated with the labels 1,3. For example, a diagram may have crossings labelled with the pairs. The Dowker notation for this labelling is the sequence,6 −1228 −4 −10, Knots tabulations typically consider only prime knots and disregard chirality, so this ambiguity does not affect the tabulation. The ménage problem, posed by Tait, concerns counting the number of different number sequences possible in this notation, conway notation Alexander–Briggs notation Adams, Colin Conrad. The Knot Book, An Elementary Introduction to the Mathematical Theory of Knots
14.
0 1 knot
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The unknot arises in the mathematical theory of knots. Intuitively, the unknot is a loop of rope without a knot in it. The unknot is also called the trivial knot, an unknot is the identity element with respect to the knot sum operation. Currently there are several well-known unknot recognition algorithms, but they are known to be inefficient or have no efficient implementation. It is not known whether many of the current invariants, such as finite type invariants, are a complete invariant of the unknot, even if they were, the problem of computing them efficiently remains. Many useful practical knots are actually the unknot, including all knots which can be tied in the bight, the Alexander-Conway polynomial and Jones polynomial of the unknot are trivial, Δ =1, ∇ =1, V =1. No other knot with 10 or fewer crossings has trivial Alexander polynomial and it is an open problem whether any non-trivial knot has the same Jones polynomial as the unknot. The knot group of the unknot is a cyclic group. Knot Unlink Unknot, The Knot Atlas
15.
4 1 knot
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In knot theory, a figure-eight knot is the unique knot with a crossing number of four. This is the smallest possible crossing number except for the unknot, the figure-eight knot is a prime knot. The name is given because tying a normal figure-eight knot in a rope and then joining the ends together, in the most natural way, gives a model of the mathematical knot. A simple parametric representation of the knot is as the set of all points where x = cos y = sin z = sin for t varying over the real numbers. The figure-eight knot is prime, alternating, rational with a value of 5/2. The figure-eight knot is also a fibered knot and this follows from other, less simple representations of the knot, It is a homogeneous closed braid, and a theorem of John Stallings shows that any closed homogeneous braid is fibered. It is the link at of a critical point of a real-polynomial map F, R4→R2. Bernard Perron found the first such F for this knot, namely, F = G, the figure-eight knot has played an important role historically in the theory of 3-manifolds. Sometime in the mid-to-late 1970s, William Thurston showed that the figure-eight was hyperbolic and this construction, new at the time, led him to many powerful results and methods. For example, he was able to show all but ten Dehn surgeries on the figure-eight knot resulted in non-Haken, non-Seifert-fibered irreducible 3-manifolds. Many more have been discovered by generalizing Thurstons construction to other knots, the figure-eight knot is also the hyperbolic knot whose complement has the smallest possible volume,2.02988. According to the work of Chun Cao and Robert Meyerhoff, from this perspective, the figure-eight knot can be considered the simplest hyperbolic knot. The figure eight knot complement is a double-cover of the Gieseking manifold, however, it is not currently known whether the figure-eight knot is the only one that achieves the bound of 10. A well-known conjecture is that the bound is 6, the symmetry between q and q −1 in the Jones polynomial reflects the fact that the figure-eight knot is achiral. Ian Agol, Bounds on exceptional Dehn filling, Geometry & Topology 4, mR1799796 Chun Cao and Robert Meyerhoff, The orientable cusped hyperbolic 3-manifolds of minimum volume, Inventiones Mathematicae,146, no. MR1869847 Marc Lackenby, Word hyperbolic Dehn surgery, Inventiones Mathematicae 140, no
16.
Alternating knot
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In knot theory, a knot or link diagram is alternating if the crossings alternate under, over, under, over, as one travels along each component of the link. A link is alternating if it has an alternating diagram, many of the knots with crossing number less than 10 are alternating. The simplest non-alternating prime knots have 8 crossings and it is conjectured that as the crossing number increases, the percentage of knots that are alternating goes to 0 exponentially quickly. Alternating links end up having an important role in theory and 3-manifold theory, due to their complements having useful and interesting geometric. This led Ralph Fox to ask, What is an alternating knot, by this he was asking what non-diagrammatic properties of the knot complement would characterize alternating knots. Various geometric and topological information is revealed in an alternating diagram, primeness and splittability of a link is easily seen from the diagram. The crossing number of a reduced, alternating diagram is the number of the knot. This last is one of the celebrated Tait conjectures, an alternating knot diagram is in one-to-one correspondence with a planar graph. Each crossing is associated with an edge and half of the components of the complement of the diagram are associated with vertices in a checker board manner. The Tait conjectures are, Any reduced diagram of a link has the fewest possible crossings. Any two reduced diagrams of the alternating knot have the same writhe. Given any two reduced alternating diagrams D1 and D2 of an oriented, prime alternating link, D1 may be transformed to D2 by means of a sequence of certain simple moves called flypes, also known as the Tait flyping conjecture. Morwen Thistlethwaite, Louis Kauffman and K. Murasugi proved the first two Tait conjectures in 1987 and Morwen Thistlethwaite and William Menasco proved the Tait flyping conjecture in 1991, thus hyperbolic volume is an invariant of many alternating links. Marc Lackenby has shown that the volume has upper and lower linear bounds as functions of the number of twist regions of a reduced, adams, The Knot Book, An elementary introduction to the mathematical theory of knots. American Mathematical Society, Providence, RI,2004, xiv+307 pp. ISBN 0-8218-3678-1 William Menasco, Closed incompressible surfaces in alternating knot and link complements. Marc Lackenby, The volume of hyperbolic alternating link complements, with an appendix by Ian Agol and Dylan Thurston. Weisstein, Eric W. Taits Knot Conjectures, celtic Knotwork to build an alternating knot from its planar graph
17.
Torus knot
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In knot theory, a torus knot is a special kind of knot that lies on the surface of an unknotted torus in R3. Similarly, a link is a link which lies on the surface of a torus in the same way. Each torus knot is specified by a pair of coprime integers p and q, a torus link arises if p and q are not coprime. A torus knot is trivial if and only if p or q is equal to 1 or −1. The simplest nontrivial example is the knot, also known as the trefoil knot. A torus knot can be rendered geometrically in multiple ways which are topologically equivalent, the convention used in this article and its figures is the following. The -torus knot winds q times around a circle in the interior of the torus, if p and q are not relatively prime, then we have a torus link with more than one component. The direction in which the strands of the wrap around the torus is also subject to differing conventions. The most common is to have the form a right-handed screw for p q >0. The -torus knot can be given by the parametrization x = r cos y = r sin z = − sin where r = cos +2 and 0 < ϕ <2 π. This lies on the surface of the torus given by 2 + z 2 =1, other parameterizations are also possible, because knots are defined up to continuous deformation. The latter generalizes smoothly to any coprime p, q satisfying p < q <2 p, a torus knot is trivial iff either p or q is equal to 1 or −1. Each nontrivial torus knot is prime and chiral, the torus knot is equivalent to the torus knot. This can be proved by moving the strands on the surface of the torus, the torus knot is the obverse of the torus knot. The torus knot is equivalent to the torus knot except for the reversed orientation, any -torus knot can be made from a closed braid with p strands. The appropriate braid word is q, the crossing number of a torus knot with p, q >0 is given by c = min. The genus of a knot with p, q >0 is g =12. The Alexander polynomial of a knot is
18.
Fibered knot
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For example, The unknot, trefoil knot, and figure-eight knot are fibered knots. The Hopf link is a fibered link, fibered knots and links arise naturally, but not exclusively, in complex algebraic geometry. For instance, each point of a complex plane curve can be described topologically as the cone on a fibered knot or link called the link of the singularity. The trefoil knot is the link of the cusp singularity z 2 + w 3, in these cases, the family of Seifert surfaces is an aspect of the Milnor fibration of the singularity. A knot is fibered if and only if it is the binding of some open book decomposition of S3, the Alexander polynomial of a fibered knot is monic, i. e. the coefficients of the highest and lowest powers of t are plus or minus 1. Examples of knots with nonmonic Alexander polynomials abound, for example the twist knots have Alexander polynomials qt − + qt−1, in particular the Stevedores knot is not fibered. Pretzel knot How to construct all fibered knots and links
19.
Pretzel link
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In the mathematical theory of knots, a pretzel link is a special kind of link. A pretzel link which is also a knot is a pretzel knot, in the standard projection of the pretzel link, there are p 1 left-handed crossings in the first tangle, p 2 in the second, and, in general, p n in the nth. A pretzel link can also be described as a Montesinos link with integer tangles, the pretzel link is a knot iff both n and all the p i are odd or exactly one of the p i is even. The pretzel link is split if at least two of the p i are zero, but the converse is false, the pretzel link is the mirror image of the pretzel link. The pretzel link is link-equivalent to the pretzel link, thus, too, the pretzel link is link-equivalent to the pretzel link. The pretzel link is link-equivalent to the pretzel link, however, if one orients the links in a canonical way, then these two links have opposite orientations. The pretzel knot is the trefoil, the knot is its mirror image. The pretzel knot is the stevedore knot, if p, q, r are distinct odd integers greater than 1, then the pretzel knot is a non-invertible knot. The pretzel link is a formed by three linked unknots. The pretzel knot is the sum of two trefoil knots. The pretzel link is the union of an unknot and another knot. A Montesinos link is a kind of link that generalizes pretzel links. A Montesinos link which is also a knot is a Montesinos knot, a Montesinos link is composed of several rational tangles. One notation for a Montesinos link is K, in this notation, e and all the α i and β i are integers. Many results have been stated about the manifolds that result from Dehn surgery on the knot in particular. The hyperbolic volume of the complement of the link is 4 times Catalans constant. This pretzel link complement is one of two two-cusped hyperbolic manifolds with the possible volume, the other being the complement of the Whitehead link. Trotter, Hale F. Non-invertible knots exist, Topology,2, 272–280
20.
Prime knot
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In knot theory, a prime knot or prime link is a knot that is, in a certain sense, indecomposable. Specifically, it is a knot which cannot be written as the knot sum of two non-trivial knots. Knots that are not prime are said to be composite knots or composite links and it can be a nontrivial problem to determine whether a given knot is prime or not. A family of examples of prime knots are the torus knots and these are formed by wrapping a circle around a torus p times in one direction and q times in the other, where p and q are coprime integers. The simplest prime knot is the trefoil with three crossings, the trefoil is actually a -torus knot. The figure-eight knot, with four crossings, is the simplest non-torus knot, for any positive integer n, there are a finite number of prime knots with n crossings. The first few values are given in the following table, enantiomorphs are counted only once in this table and the following chart. A theorem due to Horst Schubert states that every knot can be expressed as a connected sum of prime knots. List of prime knots Weisstein, Eric W, prime Links with a Non-Prime Component, The Knot Atlas
21.
Reversible knot
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In the mathematical field of knot theory, a chiral knot is a knot that is not equivalent to its mirror image. An oriented knot that is equivalent to its image is an amphichiral knot. The chirality of a knot is a knot invariant, a knots chirality can be further classified depending on whether or not it is invertible. The chirality of certain knots was long suspected, and was proven by Max Dehn in 1914, P. G. Tait conjectured that all amphichiral knots had even crossing number, but a counterexample was found by Morwen Thistlethwaite et al. in 1998. However, Taits conjecture was true for prime, alternating knots. The simplest chiral knot is the knot, which was shown to be chiral by Max Dehn. The Alexander polynomial cannot detect the chirality of a knot, but the Jones polynomial can in some cases, if Vk ≠ Vk, then the knot is chiral, however the converse is not true. The HOMFLY polynomial is even better at detecting chirality, but there is no known polynomial knot invariant which can fully detect chirality, a chiral knot that is invertible is classified as a reversible knot. If a knot is not equivalent to its inverse or its image, it is a fully chiral knot. An amphichiral knot is one which has an orientation-reversing self-homeomorphism of the 3-sphere, α, all amphichiral alternating knots have even crossing number. The first amphichiral knot with odd crossing number is a 15-crossing knot discovered by Hoste et al, if a knot is isotopic to both its reverse and its mirror image, it is fully amphichiral. The simplest knot with this property is the figure-eight knot, if the self-homeomorphism, α, preserves the orientation of the knot, it is said to be positive amphichiral. This is equivalent to the knot being isotopic to its mirror, no knots with crossing number smaller than twelve are positive amphichiral. If the self-homeomorphism, α, reverses the orientation of the knot and this is equivalent to the knot being isotopic to the reverse of its mirror image. The knot with this property that has the fewest crossings is the knot 817
22.
Tricolorable knot
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In the mathematical field of knot theory, the tricolorability of a knot is the ability of a knot to be colored with three colors subject to certain rules. Tricolorability is an invariant, and hence can be used to distinguish between two different knots. In particular, since the unknot is not tricolorable, any knot is necessarily nontrivial. A knot is tricolorable if each strand of the diagram can be colored one of three colors, subject to the following rules,1. At least two colors must be used, and 2, at each crossing, the three incident strands are either all the same color or all different colors. Some references state instead that all three colors must be used, for a knot, this is equivalent to the definition above, however, for a link it is not. The trefoil knot and trivial 2-link are tricolorable, but the unknot, Whitehead link, if the projection of a knot is tricolorable, then Reidemeister moves on the knot preserve tricolorability, so either every projection of a knot is tricolorable or none is. Here is an example of how to color a knot in accordance of the rules of tricolorability, by convention, knot theorists use the colors red, green, and blue. In this coloring the three strands at every crossing have three different colors, coloring one but not both of the trefoil knots all red would also give an admissible coloring. The true lovers knot is also tricolorable, the figure-eight knot is not tricolorable. In the diagram shown, it has four strands with each pair of strands meeting at some crossing, if three of the strands had the same color, then all strands would be forced to be the same color. Otherwise each of four strands must have a distinct color. Since tricolorability is an invariant, none of its other diagrams can be tricolored either. Tricolorability is an invariant, which is a property of a knot or link that remains constant regardless of any ambient isotopy. This can be proven by examining Reidemeister moves, since each Reidemeister move can be made without affecting tricolorability, tricolorability is an isotopy invariant. Because tricolorability is a classification, it is a relatively weak invariant. The composition of a knot with another knot is always tricolorable. A way to strengthen the invariant is to count the number of possible 3-colorings, in this case, the rule that at least two colors are used is relaxed and now every link has at least three 3-colorings
23.
Twist knot
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In knot theory, a branch of mathematics, a twist knot is a knot obtained by repeatedly twisting a closed loop and then linking the ends together. The twist knots are a family of knots, and are considered the simplest type of knots after the torus knots. A twist knot is obtained by linking together the two ends of a twisted loop, any number of half-twists may be introduced into the loop before linking, resulting in an infinite family of possibilities. The following figures show the first few twist knots, All twist knots have unknotting number one, every twist knot is also a 2-bridge knot. Of the twist knots, only the unknot and the knot are slice knots. A twist knot with n half-twists has crossing number n +2, All twist knots are invertible, but the only amphichiral twist knots are the unknot and the figure-eight knot. The invariants of a twist knot depend on the n of half-twists
24.
Topology
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In mathematics, topology is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing. This can be studied by considering a collection of subsets, called open sets, important topological properties include connectedness and compactness. Topology developed as a field of study out of geometry and set theory, through analysis of such as space, dimension. Such ideas go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs, Leonhard Eulers Seven Bridges of Königsberg Problem and Polyhedron Formula are arguably the fields first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century, by the middle of the 20th century, topology had become a major branch of mathematics. It defines the basic notions used in all branches of topology. Algebraic topology tries to measure degrees of connectivity using algebraic constructs such as homology, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to geometry and together they make up the geometric theory of differentiable manifolds. Geometric topology primarily studies manifolds and their embeddings in other manifolds, a particularly active area is low-dimensional topology, which studies manifolds of four or fewer dimensions. This includes knot theory, the study of mathematical knots, Topology, as a well-defined mathematical discipline, originates in the early part of the twentieth century, but some isolated results can be traced back several centuries. Among these are certain questions in geometry investigated by Leonhard Euler and his 1736 paper on the Seven Bridges of Königsberg is regarded as one of the first practical applications of topology. On 14 November 1750 Euler wrote to a friend that he had realised the importance of the edges of a polyhedron and this led to his polyhedron formula, V − E + F =2. Some authorities regard this analysis as the first theorem, signalling the birth of topology, further contributions were made by Augustin-Louis Cauchy, Ludwig Schläfli, Johann Benedict Listing, Bernhard Riemann and Enrico Betti. Listing introduced the term Topologie in Vorstudien zur Topologie, written in his native German, in 1847, the term topologist in the sense of a specialist in topology was used in 1905 in the magazine Spectator. Their work was corrected, consolidated and greatly extended by Henri Poincaré, in 1895 he published his ground-breaking paper on Analysis Situs, which introduced the concepts now known as homotopy and homology, which are now considered part of algebraic topology. Unifying the work on function spaces of Georg Cantor, Vito Volterra, Cesare Arzelà, Jacques Hadamard, Giulio Ascoli and others, Maurice Fréchet introduced the metric space in 1906. A metric space is now considered a case of a general topological space. In 1914, Felix Hausdorff coined the term topological space and gave the definition for what is now called a Hausdorff space, currently, a topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski
25.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
26.
Knot (mathematics)
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In mathematics, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3, considered up to continuous deformations. A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed—there are no ends to tie or untie on a mathematical knot. Physical properties such as friction and thickness also do not apply, the term knot is also applied to embeddings of S j in S n, especially in the case j = n −2. The branch of mathematics that studies knots is known as knot theory, a knot is an embedding of the circle into three-dimensional Euclidean space. Or the 3-sphere, S3, since the 3-sphere is compact, two knots are defined to be equivalent if there is an ambient isotopy between them. A knot in R3, can be projected onto a plane R2, in this case, by choosing a projection side, one can completely encode the isotopy class of the knot by its regular projection by recording a simple over/under information at these crossings. In graph theory terms, a projection of a knot. The local modifications of this graph which allow to go from one diagram to any other diagram of the knot are called Reidemeister moves. The simplest knot, called the unknot or trivial knot, is a circle embedded in R3. In the ordinary sense of the word, the unknot is not knotted at all, the simplest nontrivial knots are the trefoil knot, the figure-eight knot and the cinquefoil knot. Several knots, linked or tangled together, are called links, Knots are links with a single component. A polygonal knot is a knot whose image in R3 is the union of a set of line segments. A tame knot is any knot equivalent to a polygonal knot, Knots which are not tame are called wild, and can have pathological behavior. In knot theory and 3-manifold theory, often the adjective tame is omitted, smooth knots, for example, are always tame. A framed knot is the extension of a knot to an embedding of the solid torus D2 × S1 in S3. The framing of the knot is the number of the image of the ribbon I × S1 with the knot. A framed knot can be seen as the ribbon and the framing is the number of twists. This definition generalizes to a one for framed links
27.
Clover
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Clover or trefoil are common names for plants of the genus Trifolium, consisting of about 300 species of plants in the leguminous pea family Fabaceae. They are small annual, biennial, or short-lived perennial herbaceous plants. The leaves are trifoliate, with adnate to the leaf-stalk, and heads or dense spikes of small red, purple, white, or yellow flowers. Other closely related genera often called clovers include Melilotus and Medicago, several species of clover are extensively cultivated as fodder plants. The most widely cultivated clovers are white clover Trifolium repens and red clover Trifolium pratense, in many areas, particularly on acidic soil, clover is short-lived because of a combination of insect pests, diseases and nutrient balance, this is known as clover sickness. When crop rotations are managed so that clover does not recur at intervals shorter than eight years, clover sickness in more recent times may also be linked to pollinator decline, clovers are most efficiently pollinated by bumblebees, which have declined as a result of agricultural intensification. Honeybees can also pollinate clover, and beekeepers are often in demand from farmers with clover pastures. Farmers reap the benefits of increased reseeding that occurs with increased bee activity, beekeepers benefit from the clover bloom, as clover is one of the main nectar sources for honeybees. Trifolium repens, white or Dutch clover, is a perennial abundant in meadows, the flowers are white or pinkish, becoming brown and deflexed as the corolla fades. Trifolium hybridum, alsike or Swedish clover, is a perennial which was introduced early in the 19th century and has now become naturalized in Britain, the flowers are white or rosy, and resemble those of the last species. Trifolium medium, meadow or zigzag clover, a perennial with straggling flexuous stems, clovers occasionally have four leaflets, instead of the usual three. These four-leaf clovers, like other rarities, are considered lucky, clovers can also have five, six, or more leaflets, but these are rarer. The record for most leaflets is 56, set on 10 May 2009 and this beat the 21-leaf clover, a record set in June 2008 by the same discoverer, who had also held the prior Guinness World Record of 18. A common idiom is to be in clover, meaning to live a life of ease, comfort. The cloverleaf interchange is named for the resemblance to the leaflets of a clover when viewed from the air, quattrofolium Edibility of clover, Edible parts and visual identification of wild clover
28.
Curve
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In mathematics, a curve is, generally speaking, an object similar to a line but that need not be straight. Thus, a curve is a generalization of a line, in that curvature is not necessarily zero, various disciplines within mathematics have given the term different meanings depending on the area of study, so the precise meaning depends on context. However, many of these meanings are special instances of the definition which follows, a curve is a topological space which is locally homeomorphic to a line. In everyday language, this means that a curve is a set of points which, near each of its points, looks like a line, a simple example of a curve is the parabola, shown to the right. A large number of curves have been studied in multiple mathematical fields. A closed curve is a curve that forms a path whose starting point is also its ending point—that is, closely related meanings include the graph of a function and a two-dimensional graph. Interest in curves began long before they were the subject of mathematical study and this can be seen in numerous examples of their decorative use in art and on everyday objects dating back to prehistoric times. Curves, or at least their graphical representations, are simple to create, historically, the term line was used in place of the more modern term curve. Hence the phrases straight line and right line were used to distinguish what are called lines from curved lines. For example, in Book I of Euclids Elements, a line is defined as a breadthless length, Euclids idea of a line is perhaps clarified by the statement The extremities of a line are points. Later commentators further classified according to various schemes. For example, Composite lines Incomposite lines Determinate Indeterminate The Greek geometers had studied many kinds of curves. One reason was their interest in solving problems that could not be solved using standard compass. These curves include, The conic sections, deeply studied by Apollonius of Perga The cissoid of Diocles, studied by Diocles, the conchoid of Nicomedes, studied by Nicomedes as a method to both double the cube and to trisect an angle. The Archimedean spiral, studied by Archimedes as a method to trisect an angle, the spiric sections, sections of tori studied by Perseus as sections of cones had been studied by Apollonius. A fundamental advance in the theory of curves was the advent of analytic geometry in the seventeenth century and this enabled a curve to be described using an equation rather than an elaborate geometrical construction. Previously, curves had been described as geometrical or mechanical according to how they were, or supposedly could be, conic sections were applied in astronomy by Kepler. Newton also worked on an example in the calculus of variations
29.
Parametric equation
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In mathematics, parametric equations define a group of quantities as functions of one or more independent variables called parameters. For example, the equations x = cos t y = sin t form a representation of the unit circle. Parametric equations are used in kinematics, where the trajectory of an object is represented by equations depending on time as the parameter. Because of this application, a parameter is often labeled t, however. Parameterizations are non-unique, more than one set of equations can specify the same curve. In kinematics, objects paths through space are described as parametric curves. Used in this way, the set of equations for the objects coordinates collectively constitute a vector-valued function for position. Such parametric curves can then be integrated and differentiated termwise, thus, if a particles position is described parametrically as r = then its velocity can be found as v = r ′ = and its acceleration as a = r ″ =. Another important use of equations is in the field of computer-aided design. For example, consider the three representations, all of which are commonly used to describe planar curves. These problems can be addressed by rewriting the non-parametric equations in parametric form, numerous problems in integer geometry can be solved using parametric equations. A classical such solution is Euclids parametrization of right triangles such that the lengths of their sides a, b, by multiplying a, b and c by an arbitrary positive integer, one gets a parametrization of all right triangles whose three sides have integer lengths. Converting a set of equations to a single equation involves eliminating the variable t from the simultaneous equations x = x, y = y. If one of these equations can be solved for t, the expression obtained can be substituted into the equation to obtain an equation involving x and y only. If the parametrization is given by rational functions x = p r, y = q r, where p, q, r are set-wise coprime polynomials, in some cases there is no single equation in closed form that is equivalent to the parametric equations. The simplest equation for a parabola, y = x 2 can be parameterized by using a free parameter t, and setting x = t, y = t 2 f o r − ∞ < t < ∞. More generally, any given by an explicit equation y = f can be parameterized by using a free parameter t. A more sophisticated example is the following, consider the unit circle which is described by the ordinary equation x 2 + y 2 =1
30.
Torus
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In geometry, a torus is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle. If the axis of revolution does not touch the circle, the surface has a shape and is called a torus of revolution. Real-world examples of objects include inner tubes, swim rings, and the surface of a doughnut. A torus should not be confused with a solid torus, which is formed by rotating a disc, rather than a circle, a solid torus is a torus plus the volume inside the torus. Real-world approximations include doughnuts, vadai or vada, many lifebuoys, in topology, a ring torus is homeomorphic to the Cartesian product of two circles, S1 × S1, and the latter is taken to be the definition in that context. It is a compact 2-manifold of genus 1, the ring torus is one way to embed this space into three-dimensional Euclidean space, but another way to do this is the Cartesian product of the embedding of S1 in the plane. This produces an object called the Clifford torus, a surface in 4-space. In the field of topology, a torus is any space that is topologically equivalent to a torus. R is known as the radius and r is known as the minor radius. The ratio R divided by r is known as the aspect ratio, a doughnut has an aspect ratio of about 2 to 3. An implicit equation in Cartesian coordinates for a torus radially symmetric about the z-axis is 2 + z 2 = r 2, or the solution of f =0, algebraically eliminating the square root gives a quartic equation,2 =4 R2. The three different classes of standard tori correspond to the three aspect ratios between R and r, When R > r, the surface will be the familiar ring torus. R = r corresponds to the torus, which in effect is a torus with no hole. R < r describes the self-intersecting spindle torus, when R =0, the torus degenerates to the sphere. When R ≥ r, the interior 2 + z 2 < r 2 of this torus is diffeomorphic to a product of an Euclidean open disc, the losses in surface area and volume on the inner side of the tube exactly cancel out the gains on the outer side. In traditional spherical coordinates there are three measures, R, the distance from the center of the system, and θ and φ. As a torus has, effectively, two points, the centerpoints of the angles are moved, φ measures the same angle as it does in the spherical system. The center point of θ is moved to the center of r and these terms were first used in a discussion of the Earths magnetic field, where poloidal was used to denote the direction toward the poles
31.
Homotopy
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A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology. In practice, there are difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra. If we think of the parameter of H as time then H describes a continuous deformation of f into g, at time 0 we have the function f. We can also think of the second parameter as a control that allows us to smoothly transition from f to g as the slider moves from 0 to 1. The two versions coincide by setting ht = H and it is not sufficient to require each map ht to be continuous. The animation that is looped above right provides an example of a homotopy between two embeddings, f and g, of the torus into R3. The animation shows the image of ht as a function of the parameter t and it pauses, then shows the image as t varies back from 1 to 0, pauses, and repeats this cycle. Continuous functions f and g are said to be homotopic if, being homotopic is an equivalence relation on the set of all continuous functions from X to Y. The maps f and g are called homotopy equivalences in this case, every homeomorphism is a homotopy equivalence, but the converse is not true, for example, a solid disk is not homeomorphic to a single point, although the disk and the point are homotopy equivalent. As another example, the Möbius strip and an untwisted strip are homotopy equivalent, spaces that are homotopy equivalent to a point are called contractible. Intuitively, two spaces X and Y are homotopy equivalent if they can be transformed into one another by bending, shrinking and expanding operations, the first example of a homotopy equivalence is R n with a point, denoted R n ≃. There is an equivalence between S1 and R2 −. More generally, R n − ≃ S n −1, any fiber bundle π, E → B with fibers F b homotopy equivalent to a point has homotopy equivalent total and base spaces. This generalizes the previous two examples since π, R n − → S n −1 is a bundle with fiber R >0. Every vector bundle is a bundle with a fiber homotopy equivalent to a point. For any 0 ≤ k < n, R n − R k ≃ S n − k −1 by writing R n as R k × R n − k, a function f is said to be null-homotopic if it is homotopic to a constant function. For example, a map f from the unit circle S1 to any space X is null-homotopic precisely when it can be extended to a map from the unit disk D2 to X that agrees with f on the boundary
32.
Mirror image
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A mirror image is a reflected duplication of an object that appears almost identical, but is reversed in the direction perpendicular to the mirror surface. As an optical effect it results from reflection off of such as a mirror or water. It is also a concept in geometry and can be used as a process for 3-D structures. Two-dimensional mirror images can be seen in the reflections of mirrors or other reflecting surfaces, or on a printed surface seen inside out. If we look at an object that is effectively two-dimensional and then turn it towards a mirror, in this example, it is the change in orientation rather than the mirror itself that causes the observed reversal. Another example is when we stand with our backs to the mirror, then we compare the object with its reflection by turning ourselves 180 degrees, towards the mirror. Again we perceive a left-right reversal due to a change in orientation, so, in these examples the mirror does not actually cause the observed reversals. The concept of reflection can be extended to three-dimensional objects, including the inside parts, the term then relates to structural as well as visual aspects. A three-dimensional object is reversed in the perpendicular to the mirror surface. In physics, mirror images are investigated in the subject called geometrical optics, in chemistry, two versions of a molecule, one a mirror image of the other, are called enantiomers if they are not superposable on each other. That is an example of chirality, in general, an object and its mirror image are called enantiomorphs. If a point of an object has coordinates then the image of this point has coordinates, thus reflection is a reversal of the coordinate axis perpendicular to the mirrors surface. In everyday use, a mirror does not reverse right and left, however, there is often a perception of left-right reversal, probably because the left and right of an object are defined by its top and front. Reflection in a mirror does result in a change in chirality, as a consequence, if one looks in a mirror and lets two axes coincide with those in the mirror, then this gives a reversal of the third axis. Similarly, if you stand side-on to a mirror your left and its important to realise there are only two enantiomorphs, the object and its image. So, no matter how the object is oriented towards the mirror, all the images are fundamentally identical. In the photograph of the urn and mirror, the urn is fairly symmetrical front-back, so, its not surprising that no obvious reversal of the urn can be seen in the mirror image. A mirror image appears more obviously three-dimensional if the observer moves and this is because the relative position of objects changes as the observers perspective changes, or is different viewed with each eye
33.
Knot diagram
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In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, in mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3. Knots can be described in various ways, given a method of description, however, there may be more than one description that represents the same knot. For example, a method of describing a knot is a planar diagram called a knot diagram. Any given knot can be drawn in different ways using a knot diagram. Therefore, a problem in knot theory is determining when two descriptions represent the same knot. A complete algorithmic solution to this exists, which has unknown complexity. In practice, knots are often distinguished by using a knot invariant, important invariants include knot polynomials, knot groups, and hyperbolic invariants. The original motivation for the founders of theory was to create a table of knots and links. Over six billion knots and links have been tabulated since the beginnings of knot theory in the 19th century, to gain further insight, mathematicians have generalized the knot concept in several ways. Knots can be considered in other spaces and objects other than circles can be used. Higher-dimensional knots are n-dimensional spheres in m-dimensional Euclidean space, archaeologists have discovered that knot tying dates back to prehistoric times. Besides their uses such as recording information and tying objects together, knots have interested humans for their aesthetics, Knots appear in various forms of Chinese artwork dating from several centuries BC. The endless knot appears in Tibetan Buddhism, while the Borromean rings have made repeated appearances in different cultures, the Celtic monks who created the Book of Kells lavished entire pages with intricate Celtic knotwork. Mathematical studies of knots began in the 19th century with Gauss, in the 1860s, Lord Kelvins theory that atoms were knots in the aether led to Peter Guthrie Taits creation of the first knot tables for complete classification. Tait, in 1885, published a table of knots with up to ten crossings and this record motivated the early knot theorists, but knot theory eventually became part of the emerging subject of topology. This would be the approach to knot theory until a series of breakthroughs transformed the subject. In the late 1970s, William Thurston introduced hyperbolic geometry into the study of knots with the hyperbolization theorem, many knots were shown to be hyperbolic knots, enabling the use of geometry in defining new, powerful knot invariants
34.
Algebraic geometry
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Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, the fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. A point of the plane belongs to a curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the points, the inflection points. More advanced questions involve the topology of the curve and relations between the curves given by different equations, Algebraic geometry occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis, topology and number theory. In the 20th century, algebraic geometry split into several subareas, the mainstream of algebraic geometry is devoted to the study of the complex points of the algebraic varieties and more generally to the points with coordinates in an algebraically closed field. The study of the points of a variety with coordinates in the field of the rational numbers or in a number field became arithmetic geometry. The study of the points of an algebraic variety is the subject of real algebraic geometry. A large part of singularity theory is devoted to the singularities of algebraic varieties, with the rise of the computers, a computational algebraic geometry area has emerged, which lies at the intersection of algebraic geometry and computer algebra. It consists essentially in developing algorithms and software for studying and finding the properties of explicitly given algebraic varieties and this means that a point of such a scheme may be either a usual point or a subvariety. This approach also enables a unification of the language and the tools of algebraic geometry, mainly concerned with complex points. Wiless proof of the longstanding conjecture called Fermats last theorem is an example of the power of this approach. For instance, the sphere in three-dimensional Euclidean space R3 could be defined as the set of all points with x 2 + y 2 + z 2 −1 =0. A slanted circle in R3 can be defined as the set of all points which satisfy the two polynomial equations x 2 + y 2 + z 2 −1 =0, x + y + z =0, first we start with a field k. In classical algebraic geometry, this field was always the complex numbers C and we consider the affine space of dimension n over k, denoted An. When one fixes a system, one may identify An with kn. The purpose of not working with kn is to emphasize that one forgets the vector space structure that kn carries, the property of a function to be polynomial does not depend on the choice of a coordinate system in An. When a coordinate system is chosen, the functions on the affine n-space may be identified with the ring of polynomial functions in n variables over k
35.
3-sphere
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In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a central point in 4-dimensional Euclidean space. A 3-sphere is an example of a 3-manifold, in coordinates, a 3-sphere with center and radius r is the set of all points in real, 4-dimensional space such that ∑ i =032 =2 +2 +2 +2 = r 2. The 3-sphere centered at the origin with radius 1 is called the unit 3-sphere and is usually denoted S3 and it is often convenient to regard R4 as the space with 2 complex dimensions or the quaternions. The unit 3-sphere is then given by S3 = or S3 = and this description as the quaternions of norm one, identifies the 3-sphere with the versors in the quaternion division ring. Just as the circle is important for planar polar coordinates. See polar decomposition of a quaternion for details of development of the three-sphere. This view of the 3-sphere is the basis for the study of space as developed by Georges Lemaître. The 3-dimensional cubic hyperarea of a 3-sphere of radius r is 2 π2 r 3 while the 4-dimensional quartic hypervolume is 12 π2 r 4, every non-empty intersection of a 3-sphere with a three-dimensional hyperplane is a 2-sphere. Then the 2-sphere shrinks again down to a point as the 3-sphere leaves the hyperplane. A 3-sphere is a compact, connected, 3-dimensional manifold without boundary, what this means, in the broad sense, is that any loop, or circular path, on the 3-sphere can be continuously shrunk to a point without leaving the 3-sphere. The Poincaré conjecture, proved in 2003 by Grigori Perelman, provides that the 3-sphere is the only three-dimensional manifold with these properties, the 3-sphere is homeomorphic to the one-point compactification of R3. In general, any space that is homeomorphic to the 3-sphere is called a topological 3-sphere. The homology groups of the 3-sphere are as follows, H0, any topological space with these homology groups is known as a homology 3-sphere. Initially Poincaré conjectured that all homology 3-spheres are homeomorphic to S3, infinitely many homology spheres are now known to exist. For example, a Dehn filling with slope 1/n on any knot in the 3-sphere gives a homology sphere, as to the homotopy groups, we have π1 = π2 = and π3 is infinite cyclic. The higher-homotopy groups are all finite abelian but otherwise follow no discernible pattern, for more discussion see homotopy groups of spheres. The 3-sphere is naturally a smooth manifold, in fact, an embedded submanifold of R4
36.
Complex plane curve
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In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves, and algebraic plane curves, a smooth plane curve is a curve in a real Euclidean plane R2 and is a one-dimensional smooth manifold. This means that a plane curve is a plane curve which locally looks like a line, in the sense that near every point. An algebraic plane curve is a curve in an affine or projective plane given by one polynomial equation f =0 Algebraic curves were studied extensively since the 18th century, for example, the circle given by the equation x2 + y2 =1 has degree 2. The non-singular plane algebraic curves of degree 2 are called conic sections, the plane curves of degree 3 are called cubic plane curves and, if they are non-singular, elliptic curves. Those of degree four are called plane curves. Algebraic curve Differential geometry Algebraic geometry Plane curve fitting Projective varieties Two-dimensional graph Coolidge, a Treatise on Algebraic Plane Curves, Dover Publications, ISBN 0-486-49576-0. A handbook on curves and their properties, J. W. Edwards, ASIN B0007EKXV0
37.
Polynomial
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In mathematics, a polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents. An example of a polynomial of a single indeterminate x is x2 − 4x +7, an example in three variables is x3 + 2xyz2 − yz +1. Polynomials appear in a variety of areas of mathematics and science. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, central concepts in algebra, the word polynomial joins two diverse roots, the Greek poly, meaning many, and the Latin nomen, or name. It was derived from the binomial by replacing the Latin root bi- with the Greek poly-. The word polynomial was first used in the 17th century, the x occurring in a polynomial is commonly called either a variable or an indeterminate. When the polynomial is considered as an expression, x is a symbol which does not have any value. It is thus correct to call it an indeterminate. However, when one considers the function defined by the polynomial, then x represents the argument of the function, many authors use these two words interchangeably. It is a convention to use uppercase letters for the indeterminates. However one may use it over any domain where addition and multiplication are defined, in particular, when a is the indeterminate x, then the image of x by this function is the polynomial P itself. This equality allows writing let P be a polynomial as a shorthand for let P be a polynomial in the indeterminate x. A polynomial is an expression that can be built from constants, the word indeterminate means that x represents no particular value, although any value may be substituted for it. The mapping that associates the result of substitution to the substituted value is a function. This can be expressed concisely by using summation notation, ∑ k =0 n a k x k That is. Each term consists of the product of a number—called the coefficient of the term—and a finite number of indeterminates, because x = x1, the degree of an indeterminate without a written exponent is one. A term and a polynomial with no indeterminates are called, respectively, a constant term, the degree of a constant term and of a nonzero constant polynomial is 0. The degree of the polynomial,0, is generally treated as not defined
38.
Cuspidal cubic
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In mathematics, a cuspidal cubic or semicubical parabola is an algebraic plane curve defined parametrically as x = t 2 y = a t 3. Its implicit equation is y 2 − a 2 x 3 =0 and this cubic curve has a singular point at the origin, which is a cusp. If one sets u = at, X = a2x, and Y = a3y, a special case of the semicubical parabola is the evolute of the parabola. It has the equation x =3423 +12, expanding the Tschirnhausen cubic catacaustic shows that it is also a semicubical parabola, x =3 =3 t 2 −9 y = t = t 3 −3 t. The semicubical parabola was discovered in 1657 by William Neile who computed its arc length, although the lengths of some other non-algebraic curves including the logarithmic spiral and cycloid had already been computed, the semicubical parabola was the first algebraic curve to be rectified. The isochrone curve property of the parabola was published by James Bernoulli in 1690. OConnor, John J. Robertson, Edmund F. Neiles Semi-cubical Parabola, MacTutor History of Mathematics archive, University of St Andrews
39.
Chirality (mathematics)
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In geometry, a figure is chiral if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. An object that is not chiral is said to be achiral, in 3 dimensions, not all achiral objects have a mirror plane. For example, a 3-dimensional object with inversion centre as its only nontrivial symmetry operation is achiral but has no mirror plane, a chiral object and its mirror image are said to be enantiomorphs. The word chirality is derived from the Greek χείρ, the hand, the most familiar chiral object, a non-chiral figure is called achiral or amphichiral. Some chiral three-dimensional objects, such as the helix, can be assigned a right or left handedness, many other familiar objects exhibit the same chiral symmetry of the human body, such as gloves and shoes. A right shoe is different from a left shoe only for being mirror images of each other, in contrast thin gloves may not be considered chiral if you can wear them inside-out. The J, L, S and Z-shaped tetrominoes of the video game Tetris also exhibit chirality. Individually they contain no mirror symmetry in the plane, a figure is achiral if and only if its symmetry group contains at least one orientation-reversing isometry. In three dimensions, every figure that possesses a plane of symmetry S1, an inversion center of symmetry S2. Note, however, that there are achiral figures lacking both plane and center of symmetry, an example is the figure F0 = which is invariant under the orientation reversing isometry ↦ and thus achiral, but it has neither plane nor center of symmetry. The figure F1 = also is achiral as the origin is a center of symmetry, note also that achiral figures can have a center axis. In two dimensions, every figure which possesses an axis of symmetry is achiral, and it can be shown that every bounded achiral figure must have an axis of symmetry and its symmetry group is a frieze group generated by a single glide reflection. A knot is called if it can be continuously deformed into its mirror image. For example the unknot and the knot are achiral, whereas the trefoil knot is chiral. Chirality in Metric Spaces, Symmetry, Culture and Science 21, pp. 27–36 Chiral Polyhedra by Eric W. Weisstein, when Topology Meets Chemistry by Erica Flapan. Chiral manifold at the Manifold Atlas
40.
Tricolorability
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In the mathematical field of knot theory, the tricolorability of a knot is the ability of a knot to be colored with three colors subject to certain rules. Tricolorability is an invariant, and hence can be used to distinguish between two different knots. In particular, since the unknot is not tricolorable, any knot is necessarily nontrivial. A knot is tricolorable if each strand of the diagram can be colored one of three colors, subject to the following rules,1. At least two colors must be used, and 2, at each crossing, the three incident strands are either all the same color or all different colors. Some references state instead that all three colors must be used, for a knot, this is equivalent to the definition above, however, for a link it is not. The trefoil knot and trivial 2-link are tricolorable, but the unknot, Whitehead link, if the projection of a knot is tricolorable, then Reidemeister moves on the knot preserve tricolorability, so either every projection of a knot is tricolorable or none is. Here is an example of how to color a knot in accordance of the rules of tricolorability, by convention, knot theorists use the colors red, green, and blue. In this coloring the three strands at every crossing have three different colors, coloring one but not both of the trefoil knots all red would also give an admissible coloring. The true lovers knot is also tricolorable, the figure-eight knot is not tricolorable. In the diagram shown, it has four strands with each pair of strands meeting at some crossing, if three of the strands had the same color, then all strands would be forced to be the same color. Otherwise each of four strands must have a distinct color. Since tricolorability is an invariant, none of its other diagrams can be tricolored either. Tricolorability is an invariant, which is a property of a knot or link that remains constant regardless of any ambient isotopy. This can be proven by examining Reidemeister moves, since each Reidemeister move can be made without affecting tricolorability, tricolorability is an isotopy invariant. Because tricolorability is a classification, it is a relatively weak invariant. The composition of a knot with another knot is always tricolorable. A way to strengthen the invariant is to count the number of possible 3-colorings, in this case, the rule that at least two colors are used is relaxed and now every link has at least three 3-colorings
41.
Unknot
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The unknot arises in the mathematical theory of knots. Intuitively, the unknot is a loop of rope without a knot in it. The unknot is also called the trivial knot, an unknot is the identity element with respect to the knot sum operation. Currently there are several well-known unknot recognition algorithms, but they are known to be inefficient or have no efficient implementation. It is not known whether many of the current invariants, such as finite type invariants, are a complete invariant of the unknot, even if they were, the problem of computing them efficiently remains. Many useful practical knots are actually the unknot, including all knots which can be tied in the bight, the Alexander-Conway polynomial and Jones polynomial of the unknot are trivial, Δ =1, ∇ =1, V =1. No other knot with 10 or fewer crossings has trivial Alexander polynomial and it is an open problem whether any non-trivial knot has the same Jones polynomial as the unknot. The knot group of the unknot is a cyclic group. Knot Unlink Unknot, The Knot Atlas
42.
Reidemeister move
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In the mathematical area of knot theory, a Reidemeister move is any of three local moves on a link diagram. Each move operates on a region of the diagram and is one of three types, No other part of the diagram is involved in the picture of a move. The numbering for the types of moves corresponds to how many strands are involved, one important context in which the Reidemeister moves appear is in defining knot invariants. By demonstrating a property of a diagram which is not changed when we apply any of the Reidemeister moves. Many important invariants can be defined in this way, including the Jones polynomial, the type I move is the only move that affects the writhe of the diagram. The type III move is the one which does not change the crossing number of the diagram. The type I move affects neither the framing of the link nor the writhe of the knot diagram. Trace showed that two knot diagrams for the knot are related by using only type II and III moves if and only if they have the same writhe. Alexander Coward demonstrated that for link diagrams representing equivalent links, there is a sequence of ordered by type, first type I moves, then type II moves, type III. The moves before the type III moves increase crossing number while those after decrease crossing number, Coward & Lackenby proved the existence of an exponential tower upper bound on the number of Reidemeister moves required to pass between two diagrams of the same link. In detail, let n be the sum of the numbers of the two diagrams, then the upper bound is 222. In detail, for any such diagram with c crossings, the bound is 11. Hayashi proved there is also a bound, depending on crossing number. Media related to Reidemeister moves at Wikimedia Commons Alexander, James W. Briggs, on types of knotted curves, Ann. of Math. 9, 299–306, MR1722788 Hagge, Tobias, Every Reidemeister move is needed for each knot type,134, 295–301, doi,10. 1090/S0002-9939-05-07935-9, MR2170571 Hass, Joel, Lagarias, Jeffrey C. Hamburg,5, 24–32, doi,10. 1007/BF02952507, MR3069462 Trace, Bruce, On the Reidemeister moves of a classical knot, Proc
43.
Knot invariant
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In the mathematical field of knot theory, a knot invariant is a quantity defined for each knot which is the same for equivalent knots. The equivalence is given by ambient isotopy but can be given by homeomorphism. Some invariants are indeed numbers, but invariants can range from the simple, such as a yes/no answer, from the modern perspective, it is natural to define a knot invariant from a knot diagram. Of course, it must be unchanged under the Reidemeister moves, tricolorability is a particularly simple example. However, there are invariants which distinguish the unknot from all other knots, such as Khovanov homology, other invariants can be defined by considering some integer-valued function of knot diagrams and taking its minimum value over all possible diagrams of a given knot. Historically, many of the early knot invariants are not defined by first selecting a diagram but defined intrinsically, for example, knot genus is particularly tricky to compute, but can be effective. Some invariants associated with the knot complement include the group which is just the fundamental group of the complement. The knot quandle is also a complete invariant in this sense, by Mostow–Prasad rigidity, the hyperbolic structure on the complement of a hyperbolic link is unique, which means the hyperbolic volume is an invariant for these knots and links. Volume, and other hyperbolic invariants, have very effective. In recent years, there has been much interest in homological invariants of knots which categorify well-known invariants, heegaard Floer homology is a homology theory whose Euler characteristic is the Alexander polynomial of the knot. It has been effective in deducing new results about the classical invariants. Along a different line of study, there is a combinatorially defined cohomology theory of knots called Khovanov homology whose Euler characteristic is the Jones polynomial and this has recently been shown to be useful in obtaining bounds on slice genus whose earlier proofs required gauge theory. Khovanov and Rozansky have since defined several other related cohomology theories whose Euler characteristics recover other classical invariants, catharina Stroppel gave a representation theoretic interpretation of Khovanov homology by categorifying quantum group invariants. There is also growing interest from both knot theorists and scientists in understanding physical or geometric properties of knots and relating it to topological invariants, therefore, for knotted curves, ∮ K κ d s >4 π. An example of an invariant is ropelength, which is the amount of 1-inch diameter rope needed to realize a particular knot type. Linking number Finite type invariant Stick number Rolfsen, Dale, the Knot Book, an Elementary Introduction to the Mathematical Theory of Knots. KnotInfo, Table of Knot Invariants, Indiana. edu,09,10,18 April 2013 Invariants, The Knot Atlas
44.
Knot polynomial
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In the mathematical field of knot theory, a knot polynomial is a knot invariant in the form of a polynomial whose coefficients encode some of the properties of a given knot. The first knot polynomial, the Alexander polynomial, was introduced by James Waddell Alexander II in 1923, in the 1960s, John Conway came up with a skein relation for a version of the Alexander polynomial, usually referred to as the Alexander–Conway polynomial. The significance of this relation was not realized until the early 1980s. This led to the discovery of more knot polynomials, such as the so-called HOMFLY polynomial, soon after Jones discovery, Louis Kauffman noticed the Jones polynomial could be computed by means of a state-sum model, which involved the bracket polynomial, an invariant of framed knots. This opened up avenues of research linking knot theory and statistical mechanics, in the late 1980s, two related breakthroughs were made. Edward Witten demonstrated that the Jones polynomial, and similar Jones-type invariants, had an interpretation in Chern–Simons theory, viktor Vassiliev and Mikhail Goussarov started the theory of finite type invariants of knots. The coefficients of the previously named polynomials are known to be of finite type, in recent years, the Alexander polynomial has been shown to be related to Floer homology. The graded Euler characteristic of the knot Floer homology of Ozsváth, Alexander–Briggs notation is a notation that simply organizes knots by their crossing number. The order of Alexander–Briggs notation of knot is usually sured. Notice that Alexander polynomial and Conway polynomial can not recognize the difference of left-trefoil knot, so the same situation as granny knot and square knot, since the addition of knots in R3 is the product of knots in knot polynomials. Alexander polynomial Bracket polynomial HOMFLY polynomial Jones polynomial Kauffman polynomial skein relationship for a definition of the Alexander polynomial. Colin Adams, The Knot Book, American Mathematical Society, ISBN 0-8050-7380-9 W. B. R. Lickorish, An introduction to knot theory
45.
Alexander-Briggs notation
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In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, in mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3. Knots can be described in various ways, given a method of description, however, there may be more than one description that represents the same knot. For example, a method of describing a knot is a planar diagram called a knot diagram. Any given knot can be drawn in different ways using a knot diagram. Therefore, a problem in knot theory is determining when two descriptions represent the same knot. A complete algorithmic solution to this exists, which has unknown complexity. In practice, knots are often distinguished by using a knot invariant, important invariants include knot polynomials, knot groups, and hyperbolic invariants. The original motivation for the founders of theory was to create a table of knots and links. Over six billion knots and links have been tabulated since the beginnings of knot theory in the 19th century, to gain further insight, mathematicians have generalized the knot concept in several ways. Knots can be considered in other spaces and objects other than circles can be used. Higher-dimensional knots are n-dimensional spheres in m-dimensional Euclidean space, archaeologists have discovered that knot tying dates back to prehistoric times. Besides their uses such as recording information and tying objects together, knots have interested humans for their aesthetics, Knots appear in various forms of Chinese artwork dating from several centuries BC. The endless knot appears in Tibetan Buddhism, while the Borromean rings have made repeated appearances in different cultures, the Celtic monks who created the Book of Kells lavished entire pages with intricate Celtic knotwork. Mathematical studies of knots began in the 19th century with Gauss, in the 1860s, Lord Kelvins theory that atoms were knots in the aether led to Peter Guthrie Taits creation of the first knot tables for complete classification. Tait, in 1885, published a table of knots with up to ten crossings and this record motivated the early knot theorists, but knot theory eventually became part of the emerging subject of topology. This would be the approach to knot theory until a series of breakthroughs transformed the subject. In the late 1970s, William Thurston introduced hyperbolic geometry into the study of knots with the hyperbolization theorem, many knots were shown to be hyperbolic knots, enabling the use of geometry in defining new, powerful knot invariants
46.
Braid group
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In mathematics, the braid group on n strands, also known as the Artin braid group, is the group whose elements are equivalence classes of n-braids, and whose group operation is composition of braids. In this introduction let n =4, the generalization to other values of n will be straightforward. Consider two sets of four items lying on a table, with the items in each set being arranged in a vertical line, using four strands, each item of the first set is connected with an item of the second set so that a one-to-one correspondence results. Such a connection is called a braid, the set of all braids on four strands is denoted by B4. The above composition of braids is indeed a group operation, a good elementary introduction to braid groups is in the last half of the YouTube clip Visual Group Theory, Lecture 1. This is outlined in the article on braid theory, alternatively, one can define the braid group purely algebraically via the braid relations, keeping the pictures in mind only to guide the intuition. Braid groups were introduced explicitly by Emil Artin in 1925, although they were already implicit in Adolf Hurwitzs work on monodromy, consider the following three braids, Every braid in B4 can be written as a composition of a number of these braids and their inverses. In other words, these three braids generate the group B4, upon reaching the right end, the braid has been written as a product of the σs and their inverses. It is clear that σ1σ3 = σ3σ1, while the two relations are not quite as obvious, σ1σ2σ1 = σ2σ1σ2, σ2σ3σ2 = σ3σ2σ3. It can be shown that all other relations among the braids σ1, σ2 and σ3 already follow from these relations and this presentation leads to generalisations of braid groups called Artin groups. The cubic relations, known as the relations, play an important role in the theory of Yang–Baxter equation. The braid group B1 is trivial, B2 is a cyclic group Z. The n-strand braid group Bn embeds as a subgroup into the braid group Bn+1 by adding an extra strand that does not cross any of the first n strands. The increasing union of the groups with all n ≥1 is the infinite braid group B∞. All non-identity elements of Bn have infinite order, i. e. Bn is torsion-free, there is a left-invariant linear order on Bn called the Dehornoy order. For n ≥3, Bn contains an isomorphic to the free group on two generators. There is a homomorphism Bn → Z defined by σi ↦1, so for instance, the braid σ2σ3σ1−1σ2σ3 is mapped to 1 +1 −1 +1 +1 =3. This map corresponds to the abelianization of the braid group, by forgetting how the strands twist and cross, every braid on n strands determines a permutation on n elements
47.
Fiber bundle
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In mathematics, and particularly topology, a fiber bundle is a space that is locally a product space, but globally may have a different topological structure. The map π, called the projection or submersion of the bundle, is regarded as part of the structure of the bundle, the space E is known as the total space of the fiber bundle, B as the base space, and F the fiber. In the trivial case, E is just B × F, and this is called a trivial bundle. Examples of non-trivial fiber bundles include the Möbius strip and Klein bottle, Fiber bundles such as the tangent bundle of a manifold and more general vector bundles play an important role in differential geometry and differential topology, as do principal bundles. Mappings between total spaces of bundles that commute with the projection maps are known as bundle maps. A bundle map from the space itself to E is called a section of E. Fiber bundles became their own object of study in the period 1935-1940, the first general definition appeared in the works of Hassler Whitney. Whitney came to the definition of a fiber bundle from his study of a more particular notion of a sphere bundle. A fiber bundle is a structure, where E, B, the space B is called the base space of the bundle, E the total space, and F the fiber. The map π is called the projection map and we shall assume in what follows that the base space B is connected. That is, the diagram should commute, where proj1, U × F → U is the natural projection and φ. The set of all is called a trivialization of the bundle. Thus for any p in B, the preimage π−1 is homeomorphic to F and is called the fiber over p, every fiber bundle π, E → B is an open map, since projections of products are open maps. Therefore B carries the quotient topology determined by the map π, a fiber bundle is often denoted that, in analogy with a short exact sequence, indicates which space is the fiber, total space and base space, as well as the map from total to base space. A smooth fiber bundle is a bundle in the category of smooth manifolds. That is, E, B, and F are required to be smooth manifolds, let E = B × F and let π, E → B be the projection onto the first factor. Then E is a fiber bundle over B, here E is not just locally a product but globally one. Any such fiber bundle is called a trivial bundle, any fiber bundle over a contractible CW-complex is trivial
48.
Circle
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A circle is a simple closed shape in Euclidean geometry. The distance between any of the points and the centre is called the radius, a circle is a simple closed curve which divides the plane into two regions, an interior and an exterior. Annulus, the object, the region bounded by two concentric circles. Arc, any connected part of the circle, centre, the point equidistant from the points on the circle. Chord, a segment whose endpoints lie on the circle. Circumference, the length of one circuit along the circle, or the distance around the circle and it is a special case of a chord, namely the longest chord, and it is twice the radius. Disc, the region of the bounded by a circle. Lens, the intersection of two discs, passant, a coplanar straight line that does not touch the circle. Radius, a line segment joining the centre of the circle to any point on the circle itself, or the length of such a segment, sector, a region bounded by two radii and an arc lying between the radii. Segment, a region, not containing the centre, bounded by a chord, secant, an extended chord, a coplanar straight line cutting the circle at two points. Semicircle, an arc that extends from one of a diameters endpoints to the other, in non-technical common usage it may mean the diameter, arc, and its interior, a two dimensional region, that is technically called a half-disc. A half-disc is a case of a segment, namely the largest one. Tangent, a straight line that touches the circle at a single point. The word circle derives from the Greek κίρκος/κύκλος, itself a metathesis of the Homeric Greek κρίκος, the origins of the words circus and circuit are closely related. The circle has been known since before the beginning of recorded history, natural circles would have been observed, such as the Moon, Sun, and a short plant stalk blowing in the wind on sand, which forms a circle shape in the sand. The circle is the basis for the wheel, which, with related inventions such as gears, in mathematics, the study of the circle has helped inspire the development of geometry, astronomy and calculus. Some highlights in the history of the circle are,1700 BCE – The Rhind papyrus gives a method to find the area of a circular field. The result corresponds to 256/81 as a value of π.300 BCE – Book 3 of Euclids Elements deals with the properties of circles
49.
Complex number
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A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, satisfying the equation i2 = −1. In this expression, a is the part and b is the imaginary part of the complex number. If z = a + b i, then ℜ z = a, ℑ z = b, Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point in the complex plane, a complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way, the numbers are a field extension of the ordinary real numbers. As well as their use within mathematics, complex numbers have applications in many fields, including physics, chemistry, biology, economics, electrical engineering. The Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers and he called them fictitious during his attempts to find solutions to cubic equations in the 16th century. Complex numbers allow solutions to equations that have no solutions in real numbers. For example, the equation 2 = −9 has no real solution, Complex numbers provide a solution to this problem. The idea is to extend the real numbers with the unit i where i2 = −1. According to the theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. A complex number is a number of the form a + bi, for example, −3.5 + 2i is a complex number. The real number a is called the part of the complex number a + bi. By this convention the imaginary part does not include the unit, hence b. The real part of a number z is denoted by Re or ℜ. For example, Re = −3.5 Im =2, hence, in terms of its real and imaginary parts, a complex number z is equal to Re + Im ⋅ i. This expression is known as the Cartesian form of z. A real number a can be regarded as a number a + 0i whose imaginary part is 0
50.
Fibration
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In topology, a branch of mathematics, a fibration is a generalization of the notion of a fiber bundle. A fiber bundle makes precise the idea of one topological space being parameterized by another topological space, a fibration is like a fiber bundle, except that the fibers need not be the same space, nor even homeomorphic, rather, they are just homotopy equivalent. Weak fibrations discard even this equivalence for a technical property. Fiber bundles have a particularly simple homotopy theory that allows information about the bundle to be inferred from information about one or both of these constituent spaces. A fibration satisfies an additional condition guaranteeing that it will behave like a bundle from the point of view of homotopy theory. Fibrations are dual to cofibrations, with a correspondingly dual notion of the extension property. A fibration is a mapping p, E → B satisfying the homotopy lifting property with respect to any space. In homotopy theory, any mapping is as good as a fibration—i. e, any map can be decomposed as a homotopy equivalence into a mapping path space followed by a fibration into homotopy fibers. The fibers are by definition the subspaces of E that are the images of points b of B. If the base space B is path connected, it is a consequence of the definition that the fibers of two different points b1 and b2 in B are homotopy equivalent, therefore one usually speaks of the fiber F. This thesis firmly established in algebraic topology the use of spectral sequences, because a sheaf can be considered a local homeomorphism, the notions were closely interlinked at the time. One of the desirable properties of the Serre spectral sequence is to account for the action of the fundamental group of the base B on the homology of the total space E. Note that Serre fibrations are strictly weaker than fibrations in general, the lifting property need only hold on cubes. As a result, the fibers might not even be homotopy equivalent, an explicit example is given below. In the following examples, a fibration is denoted F → E → B, where the first map is the inclusion of the fiber F into the total space E and this is also referred to as a fibration sequence. The projection map from a space is very easily seen to be a fibration. Fiber bundles have local trivializations, i. e. Cartesian product structures exist locally on B, more precisely, if there are local trivializations over a numerable open cover of B, the bundle is a fibration. Any open cover of a space has a numerable refinement
51.
Seifert fiber space
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A Seifert fiber space is a 3-manifold together with a nice decomposition as a disjoint union of circles. In other words, it is a S1 -bundle over a 2-dimensional orbifold, most small 3-manifolds are Seifert fiber spaces, and they account for all compact oriented manifolds in 6 of the 8 Thurston geometries of the geometrization conjecture. A Seifert manifold is a closed 3-manifold together with a decomposition into a disjoint union of such that each fiber has a tubular neighborhood that forms a standard fibered torus. A standard fibered torus corresponding to a pair of integers with a>0 is the surface bundle of the automorphism of a disk given by rotation by an angle of 2πb/a. If a =1 the middle fiber is called ordinary, while if a>1 the middle fiber is called exceptional, a compact Seifert fiber space has only a finite number of exceptional fibers. The set of forms a 2-dimensional orbifold, denoted by B. It has an underlying 2-dimensional surface B0, but may have some special orbifold points corresponding to the exceptional fibers, the definition of Seifert fibration can be generalized in several ways. The Seifert manifold is allowed to have a boundary. In both of these cases, the base B of the fibration usually has a non-empty boundary, Seifert classified all closed Seifert fibrations in terms of the following invariants. Seifert manifolds are denoted by symbols where, ε is one of the 6 symbols, o 1, o 2, n 1, n 2, n 3, n 4, meaning, o1 if B is orientable and M is orientable. O2 if B is orientable and M is not orientable, n1 if B is not orientable and M is not orientable and all generators of π1 preserve orientation of the fiber. N2 if B is not orientable and M is orientable, so all generators of π1 reverse orientation of the fiber, n3 if B is not orientable and M is not orientable and g≥2 and exactly one generator of π1 preserves orientation of the fiber. N4 if B is not orientable and M is not orientable and g≥3, G is the genus of the underlying 2-manifold of the orbit surface. They are normalized so that 0<bi<ai when M is orientable, the Seifert fibration of the symbol can be constructed from that of symbol by using surgery to add fibers of types b and bi/ai. If we drop the normalization conditions then the symbol can be changed as follows, adding 1 to b and subtracting ai from bi has no effect. (In other words, we can add integers to each of the rational numbers If the manifold is not orientable, adding a fiber of type has no effect. Every symbol is equivalent under these operations to a unique normalized symbol, when working with unnormalized symbols, the integer b can be set to zero by adding a fiber of type. Two closed Seifert oriented or non-orientable fibrations are isomorphic as oriented or non-orientable fibrations if, the sum b + Σbi/ai is an invariant of oriented fibrations, which is zero if and only if the fibration becomes trivial after taking a finite cover of B
52.
Alexander polynomial
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In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered this, the first knot polynomial, soon after Conways reworking of the Alexander polynomial, it was realized that a similar skein relation was exhibited in Alexanders paper on his polynomial. Let K be a knot in the 3-sphere, let X be the infinite cyclic cover of the knot complement of K. This covering can be obtained by cutting the knot complement along a Seifert surface of K, there is a covering transformation t acting on X. Consider the first homology of X, denoted H1, the transformation t acts on the homology and so we can consider H1 a module over Z. This is called the Alexander invariant or Alexander module, the module is finitely presentable, a presentation matrix for this module is called the Alexander matrix. If r > s, set the equal to 0. If the Alexander ideal is principal, take a generator, this is called an Alexander polynomial of the knot, since this is only unique up to multiplication by the Laurent monomial ± t n, one often fixes a particular unique form. Alexanders choice of normalization is to make the polynomial have a constant term. Alexander proved that the Alexander ideal is nonzero and always principal, thus an Alexander polynomial always exists, and is clearly a knot invariant, denoted Δ K. The Alexander polynomial for the knot configured by only one string is a polynomial of t2, namely, it can not distinguish between the knot and one for its mirror image. The following procedure for computing the Alexander polynomial was given by J. W. Alexander in his paper, take an oriented diagram of the knot with n crossings, there are n +2 regions of the knot diagram. To work out the Alexander polynomial, first one must create a matrix of size. The n rows correspond to the n crossings, and the n +2 columns to the regions, the values for the matrix entries are either 0,1, −1, t, −t. Consider the entry corresponding to a region and crossing. If the region is not adjacent to the crossing, the entry is 0, if the region is adjacent to the crossing, the entry depends on its location. The following table gives the entry, determined by the location of the region at the crossing from the perspective of the incoming undercrossing line, depending on the columns removed, the answer will differ by multiplication by ± t n. To resolve this ambiguity, divide out the largest possible power of t and multiply by −1 if necessary, the Alexander polynomial can also be computed from the Seifert matrix
53.
Jones polynomial
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In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. Specifically, it is an invariant of a knot or link which assigns to each oriented knot or link a Laurent polynomial in the variable t 1 /2 with integer coefficients. Suppose we have an oriented link L, given as a knot diagram and we will define the Jones polynomial, V, using Kauffmans bracket polynomial, which we denote by ⟨ ⟩. Note that here the bracket polynomial is a Laurent polynomial in the variable A with integer coefficients, first, we define the auxiliary polynomial X = − w ⟨ L ⟩, where w denotes the writhe of L in its given diagram. The writhe of a diagram is the number of positive crossings minus the number of negative crossings, the writhe is not a knot invariant. X is a knot invariant since it is invariant under changes of the diagram of L by the three Reidemeister moves, invariance under type II and III Reidemeister moves follows from invariance of the bracket under those moves. The bracket polynomial is known to change by multiplication by − A ±3 under a type I Reidemeister move, the definition of the X polynomial given above is designed to nullify this change, since the writhe changes appropriately by +1 or -1 under type I moves. Now make the substitution A = t −1 /4 in X to get the Jones polynomial V and this results in a Laurent polynomial with integer coefficients in the variable t 1 /2. This construction of the Jones Polynomial for tangles is a generalization of the Kauffman bracket of a link. The construction was developed by Professor Vladimir G. Turaev and published in 1990 in the Journal of Mathematics and Science. Let k be an integer and S k denote the set of all isotopic types of tangle diagrams, with 2 k ends, having no crossing points. Jones original formulation of his polynomial came from his study of operator algebras, a theorem of Alexanders states that it is the trace closure of a braid, say with n strands. Now define a representation ρ of the group on n strands, Bn. The standard braid generator σ i is sent to A ⋅ e i + A −1 ⋅1 and it can be checked easily that this defines a representation. Take the braid word σ obtained previously from L and compute δ n −1 t r ρ where tr is the Markov trace and this gives ⟨ L ⟩, where ⟨ ⟩ is the bracket polynomial. This can be seen by considering, as Kauffman did, the Temperley–Lieb algebra as a particular diagram algebra, an advantage of this approach is that one can pick similar representations into other algebras, such as the R-matrix representations, leading to generalized Jones invariants. Thus, a knot, a knot equivalent to its mirror image, has palindromic entries in its Jones polynomial. See the article on skein relation for an example of a computation using these relations, another remarkable property of this invariant states that the Jones Polynomial of an alternating link is an alternating polynomial
54.
Motif (visual arts)
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In art and iconography, a motif is an element of an image. A motif may be repeated in a pattern or design, often many times, the related motif of confronted animals is often seen alone, but may also be repeated, for example in Byzantine silk and other ancient textiles. Ornamental or decorative art can usually be analysed into a number of different elements and these may often, as in textile art, be repeated many times in a pattern. Important examples in Western art include acanthus, egg and dart, many designs in Islamic culture are motifs, including those of the sun, moon, animals such as horses and lions, flowers, and landscapes. Motifs can have effects and be used for propaganda. The idea of a motif has become used more broadly in discussing literature, geometric, typically repeated, Meander, palmette, rosette, gul in Oriental rugs, acanthus, egg and dart, Bead and reel, Pakudos, Sauwastika, Adinkra symbols. Figurative, Master of Animals, confronted animals, velificatio, Death, dover Publications, ISBN 0-486-26869-1 Jones, Owen. Dover Publications, Revised edition, ISBN 0-486-25463-1 Welch, Patricia Bjaaland, chinese art, a guide to motifs and visual imagery. Turtle Publishing, ISBN 0-8048-3864-X Visual motifs Theater of Drawing
55.
Iconography
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The word iconography comes from the Greek εἰκών and γράφειν. A secondary meaning is the production of images, called icons, in the Byzantine and Orthodox Christian tradition. In art history, an iconography may also mean a depiction of a subject in terms of the content of the image, such as the number of figures used, their placing. Sometimes distinctions have been made between iconology and iconography, although the definitions, and so the distinction made, varies, when referring to movies, genres are immediately recognizable through their iconography, motifs that become associated with a specific genre through repetition. Gian Pietro Bellori, a 17th-century biographer of artists of his own time, describes and analyses, not always correctly, many works. Lessings study of the classical figure Amor with a torch was an early attempt to use a study of a type of image to explain the culture it originated in. These early contributions paved the way for encyclopedias, manuals, mâles lArt religieux du XIIIe siècle en France translated into English as The Gothic Image, Religious Art in France of the Thirteenth Century has remained continuously in print. In the United States, to which Panofsky immigrated in 1931, students such as Frederick Hartt, the period from 1940 can be seen as one where iconography was especially prominent in art history. These are now being digitised and made online, usually on a restricted basis. For example, the Iconclass code 71H7131 is for the subject of Bathsheba with Davids letter, whereas 71 is the whole Old Testament and 71H the story of David. A number of collections of different types have been classified using Iconclass, notably types of old master print, the collections of the Gemäldegalerie, Berlin. These are available, usually on-line or on DVD, the system can also be used outside pure art history, for example on sites like Flickr. Central to the iconography and hagiography of Indian religions are mudra or gestures with specific meanings, the symbolic use of colour to denote the Classical Elements or Mahabhuta and letters and bija syllables from sacred alphabetic scripts are other features. Under the influence of art developed esoteric meanings, accessible only to initiates. The art of Indian Religions esp, for example, Narasimha an incarnation of Vishnu though considered a wrathful deity but in few contexts is depicted in pacified mood. Conversely, in Hindu art, narrative scenes have become more common in recent centuries, especially in miniature paintings of the lives of Krishna. Having Veena in hands to denote necessity of resonating with what you going to learn, holding pear garland to indicate concentration is required to learn about some thing. A book on other hand to indicate one should relay on authentic source of learning, dressed in white colour cloth to indicate a learner should not having any own colour but inclusive to all colours
56.
Visual arts
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The visual arts are art forms such as ceramics, drawing, painting, sculpture, printmaking, design, crafts, photography, video, filmmaking, literature, and architecture. Many artistic disciplines involve aspects of the arts as well as arts of other types. Also included within the arts are the applied arts such as industrial design, graphic design, fashion design, interior design. Current usage of the visual arts includes fine art as well as the applied, decorative arts and crafts. The distinction was emphasized by artists of the Arts and Crafts Movement, Art schools made a distinction between the fine arts and the crafts, maintaining that a craftsperson could not be considered a practitioner of the arts. The increasing tendency to painting, and to a lesser degree sculpture. The Western hierarchy of genres reflected similar attitudes, training in the visual arts has generally been through variations of the apprentice and workshop systems. Visual arts have now become a subject in most education systems. Drawing is a means of making an image, using any of a variety of tools. Digital tools that simulate the effects of these are also used, the main techniques used in drawing are, line drawing, hatching, crosshatching, random hatching, scribbling, stippling, and blending. An artist who excels in drawing is referred to as a draftsman or draughtsman, drawing goes back at least 16,000 years to Paleolithic cave representations of animals such as those at Lascaux in France and Altamira in Spain. In ancient Egypt, ink drawings on papyrus, often depicting people, were used as models for painting or sculpture, drawings on Greek vases, initially geometric, later developed to the human form with black-figure pottery during the 7th century BC. Painting taken literally is the practice of applying pigment suspended in a carrier, like drawing, painting has its documented origins in caves and on rock faces. The finest examples, believed by some to be 32,000 years old, are in the Chauvet, in shades of red, brown, yellow and black, the paintings on the walls and ceilings are of bison, cattle, horses and deer. Paintings of human figures can be found in the tombs of ancient Egypt, in the great temple of Ramses II, Nefertari, his queen, is depicted being led by Isis. The Greeks contributed to painting but much of their work has been lost, one of the best remaining representations are the hellenistic Fayum mummy portraits. Another example is mosaic of the Battle of Issus at Pompeii, Greek and Roman art contributed to Byzantine art in the 4th century BC, which initiated a tradition in icon painting. Apart from the manuscripts produced by monks during the Middle Ages
57.
Triquetra
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Triquetra originally meant triangle and was used to refer to various three-cornered shapes. It has come to refer exclusively to a more complicated shape formed of three vesicae piscis, sometimes with an added circle in or around the three lobes. Also known as a trinity knot when parallel doubled-lines are in the graph and it is similar to Odins symbol, the valknut. The triquetra is found in insular art, most notably metal work. It is also found in similar artwork on Celtic crosses and slabs from the early Christian period, the fact that the triquetra rarely stood alone in medieval Celtic art has cast reasonable doubt on its use as a primary symbol of belief. In manuscripts it is used primarily as a filler or ornament in much more complex compositions. This widely recognised knot has been used as a symbol for the past two centuries by Celtic Christians, pagans and agnostics as a sign of special things and people that are threefold. The triquetra has been a symbol in Japan called Musubi Mitsugashiwa. The triquetra has been found on runestones in Northern Europe and on early Germanic coins and it presumably had pagan religious meaning and it bears a resemblance to the valknut, a symbol associated with Odin. The symbol has been used in Christian tradition as a sign of the Trinity, when modern designers began to display the triquetra as a stand-alone design, it recalled the three-leafed shamrock which was similarly offered as a Trinity symbol by Saint Patrick. Have also suggested that the triquetra has a similarity to the Christian Ιχθυς symbol, the triquetra has been used extensively on Christian sculpture, vestments, book arts and stained glass. It has been used on the page and binding of some editions of the New King James Version. A very common representation of the symbol is with a circle that goes through the three interconnected loops of the triquetra, the circle emphasises the unity of the whole combination of three forces. It is also said to symbolise Gods love around the Holy Trinity, in contemporary Ireland, it is traditional for lovers to exchange jewelry such as a necklace or rings signifying their affection. The triquetra, also known as a trinity knot, is found as a design element is popular Irish jewelry such as claddaghs. The 7th century AD, but the triquetra is the simplest possible knot, Celtic pagans, or even neopagans who are not of a Celtic cultural orientation, may use the triquetra to symbolise a variety of concepts and mythological figures. Germanic neopagan groups who use the triquetra to symbolise their faith generally believe it is originally of Norse, the symbol is also sometimes used by wiccans and some new agers to symbolise the Triple Goddess, or as a protective symbol. The triquetra is often used artistically as an element when Celtic knotwork is used
58.
Valknut
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The valknut is a symbol consisting of three interlocked triangles. It appears on a variety of objects from the record of the ancient Germanic peoples. The compound noun valknut is from the modern era, the term used for the symbol during its historical employment is unknown. The valknut receives sporadic use in popular culture and is again associated with Germanic paganism by way of its modern day revival. The valknut appears on a variety of objects found in areas inhabited by the Germanic peoples. The symbol is featured on the Nene River Ring, an Anglo-Saxon gold finger ring dated to around the 8th to 9th centuries. Additionally, the valknut appears prominently on two stones from Gotland, Sweden, the Stora Hammars I stone and the Tängelgårda stone. The historically attested instances of the symbol appear in two traditional, topologically distinct, forms, the symbol appears in unicursal form, topologically a trefoil knot also seen in the triquetra. This unicursal form is found, for example, on the Tängelgårda stone, the symbol also appears in tricursal form, consisting of three linked triangles, topologically equivalent to the Borromean rings. This tricursal form can be seen on one of the Stora Hammars stones|Stora Hammars stone I, as well as upon the Nene River Ring, although other forms are topologically possible, these are the only attested forms found so far. In Norwegian Bokmål, the term valknute is used for a polygon with a loop on each of its corners, in the English language the looped four-cornered symbol is called Saint Johns Arms. It was made of stone with three sharp-pointed corners just like the carved symbol hrungnishjarta. Comparisons have been made between this symbol description and the known as the valknut. This is thought to symbolize the power of the god to bind and unbind, mentioned in the poems, davidson says that similar symbols are found beside figures of wolves and ravens on certain cremation urns from Anglo-Saxon cemeteries in East Anglia. The Deutscher Fußball-Bund has used a logo for the German national football team inspired by the form of the valknut. The symbol is also used as a logo for the Tau Kappa Epsilon Fraternity. The symbol appears as the inlay on some of Arch Enemy/Carcass guitarist Michael Amotts signature Dean Guitars Tyrant models. It is also used as a logo by American engineering firm RedViking, the symbol, like many others associated with Germanic paganism, is used by some white supremacists but has not been classified as generally objectionable by, for example, the German government
59.
Celtic cross
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The Celtic cross is a form of Christian cross featuring a nimbus or ring that emerged in Ireland and Britain in the Early Middle Ages. A staple of Insular art, the Celtic cross is essentially a Latin cross with a nimbus surrounding the intersection of the arms, scholars have debated its exact origins, but it is related to earlier crosses featuring rings. The form gained new popularity during the Celtic Revival of the 19th century, the shape, usually decorated with interlace and other motifs from Insular art, became popular for funerary monuments and other uses, and has remained so, spreading well beyond Ireland. Ringed crosses similar to older Continental forms appeared in Ireland and Scotland in incised stone slab artwork, however, the shape achieved its greatest popularity by its use in the monumental stone high crosses, a distinctive and widespread form of Insular art. These monuments, which first appeared in the 9th century, usually take the form of a cross on a stepped or pyramidal base. The form has obvious advantages, reducing the length of unsupported side arms. There are a number of theories as to its origin in Ireland, some scholars consider the ring a holdover from earlier wooden crosses, which may have required struts to support the crossarm. Others have seen it as deriving from indigenous Bronze Age art featuring a wheel or disc around a head, however, Michael W. Herren, Shirley Ann Brown, and others believe it originates in earlier ringed crosses in Christian art. Crosses with a representing the celestial sphere developed from the writings of the Church Fathers. The cosmological cross is an important motif in Coelius Seduliuss poem Carmen Paschale and it is not clear where the first high crosses originated. The first examples date to the about the 9th century and occur in two groups, at Ahenny in Ireland, and at Iona, an Irish monastery off the Scottish coast, the Ahenny group is generally earlier. A variety of crosses bear inscriptions in ogham, an early medieval Irish alphabet, standing crosses in Ireland and areas under Irish influence tend to be shorter and more massive than their Anglo-Saxon equivalents, which have mostly lost their headpieces. Surviving, free-standing crosses are located in Cornwall, including St Pirans cross at Perranporth, Most examples in Britain were destroyed during the Protestant Reformation. By about A. D.1200 the initial wave of building came to an end in Ireland. Popular legend in Ireland says that the Christian cross was introduced by Saint Patrick or possibly Saint Declan and it has often been claimed that Patrick combined the symbol of Christianity with the sun cross to give pagan followers an idea of the importance of the cross. By linking it with the idea of the properties of the sun. Other interpretations claim that placing the cross on top of the circle represents Christs supremacy over the pagan sun, martins Cross at Iona Abbey The Celtic Revival of the mid-19th century led to an increased use and creation of Celtic crosses in Ireland. In 1853, casts of several high crosses were exhibited at the Dublin Industrial Exhibition
60.
M. C. Escher
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Maurits Cornelis Escher, or commonly M. C. Escher, was a Dutch graphic artist who made mathematically inspired woodcuts, lithographs, early in his career, he drew inspiration from nature, making studies of insects, landscapes, and plants such as lichens, all of which he used as details in his artworks. Apart from being used in a variety of papers, his work has appeared on the covers of many books. He was one of the inspirations of Douglas Hofstadters 1979 book Gödel, Escher. Maurits Cornelis Escher was born on 17 June 1898 in Leeuwarden, Friesland and he was the youngest son of the civil engineer George Arnold Escher and his second wife, Sara Gleichman. In 1903, the moved to Arnhem, where he attended primary and secondary school until 1918. Known to his friends and family as Mauk, he was a sickly child, although he excelled at drawing, his grades were generally poor. He also took carpentry and piano lessons until he was thirteen years old, in 1918, he went to the Technical College of Delft. From 1919 to 1922, Escher attended the Haarlem School of Architecture and Decorative Arts, learning drawing and he briefly studied architecture, but he failed a number of subjects and switched to decorative arts, studying under the graphic artist Samuel Jessurun de Mesquita. In 1922, an important year of his life, Escher traveled through Italy, visiting Florence, San Gimignano, Volterra, Siena, in the same year he traveled through Spain, visiting Madrid, Toledo, and Granada. He was impressed by the Italian countryside, and in Granada by the Moorish architecture of the fourteenth-century Alhambra, Escher returned to Italy, and lived in Rome from 1923 to 1935. While in Italy, Escher met Jetta Umiker – a Swiss woman, the couple settled in Rome where their first son, Giorgio Arnaldo Escher, named after his grandfather, was born. Escher and Jetta later had two sons, Arthur and Jan. He travelled frequently, visiting Viterbo in 1926, the Abruzzi in 1927 and 1929, Corsica in 1928 and 1933, Calabria in 1930, the townscapes and landscapes of these places feature prominently in his artworks. In May and June 1936, Escher travelled back to Spain, revisiting the Alhambra, the sketches he made in the Alhambra formed a major source for his work from that time on. He also studied the architecture of the Mezquita, the Moorish mosque of Cordoba and this turned out to be the last of his long study journeys, after 1937, his artworks were created in his studio rather than in the field. All the same, even his work already shows his interest in the nature of space, the unusual, perspective. In 1935, the climate in Italy became unacceptable to Escher
61.
Figure-eight knot (mathematics)
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In knot theory, a figure-eight knot is the unique knot with a crossing number of four. This is the smallest possible crossing number except for the unknot, the figure-eight knot is a prime knot. The name is given because tying a normal figure-eight knot in a rope and then joining the ends together, in the most natural way, gives a model of the mathematical knot. A simple parametric representation of the knot is as the set of all points where x = cos y = sin z = sin for t varying over the real numbers. The figure-eight knot is prime, alternating, rational with a value of 5/2. The figure-eight knot is also a fibered knot and this follows from other, less simple representations of the knot, It is a homogeneous closed braid, and a theorem of John Stallings shows that any closed homogeneous braid is fibered. It is the link at of a critical point of a real-polynomial map F, R4→R2. Bernard Perron found the first such F for this knot, namely, F = G, the figure-eight knot has played an important role historically in the theory of 3-manifolds. Sometime in the mid-to-late 1970s, William Thurston showed that the figure-eight was hyperbolic and this construction, new at the time, led him to many powerful results and methods. For example, he was able to show all but ten Dehn surgeries on the figure-eight knot resulted in non-Haken, non-Seifert-fibered irreducible 3-manifolds. Many more have been discovered by generalizing Thurstons construction to other knots, the figure-eight knot is also the hyperbolic knot whose complement has the smallest possible volume,2.02988. According to the work of Chun Cao and Robert Meyerhoff, from this perspective, the figure-eight knot can be considered the simplest hyperbolic knot. The figure eight knot complement is a double-cover of the Gieseking manifold, however, it is not currently known whether the figure-eight knot is the only one that achieves the bound of 10. A well-known conjecture is that the bound is 6, the symmetry between q and q −1 in the Jones polynomial reflects the fact that the figure-eight knot is achiral. Ian Agol, Bounds on exceptional Dehn filling, Geometry & Topology 4, mR1799796 Chun Cao and Robert Meyerhoff, The orientable cusped hyperbolic 3-manifolds of minimum volume, Inventiones Mathematicae,146, no. MR1869847 Marc Lackenby, Word hyperbolic Dehn surgery, Inventiones Mathematicae 140, no
62.
Cinquefoil knot
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In knot theory, the cinquefoil knot, also known as Solomons seal knot or the pentafoil knot, is one of two knots with crossing number five, the other being the three-twist knot. It is listed as the 51 knot in the Alexander-Briggs notation, the cinquefoil is the closed version of the double overhand knot. The cinquefoil is a prime knot and its writhe is 5, and it is invertible but not amphichiral. These are the same as the Alexander, Conway, and Jones polynomials of the knot 10132, however, the Kauffman polynomial can be used to distinguish between these two knots. The name “cinquefoil” comes from the flowers of plants in the genus Potentilla. Pentagram Trefoil knot 7₁ knot Skein relation A Pentafoil Knot at the Wayback Machine
63.
Gordian Knot
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The Gordian Knot is a legend of Phrygian Gordium associated with Alexander the Great. It is often used as a metaphor for a problem solved easily by loophole or thinking outside the box. An oracle at Telmissus decreed that the man to enter the city driving an ox-cart should become their king. A peasant farmer named Gordias drove into town on an ox-cart and his position had also been predicted earlier by an eagle landing on his cart, a sign to him from the gods, and, on entering the city, Gordias was declared king. Out of gratitude, his son Midas dedicated the ox-cart to the Phrygian god Sabazios, so, in 333 BCE, while wintering at Gordium, Alexander the Great attempted to untie the knot. When he could not find the end to the knot to unbind it, he sliced it in half with a stroke of his sword and that night there was a violent thunderstorm. Alexanders prophet Aristander took this as a sign that Zeus was pleased, once Alexander had sliced the knot with a sword-stroke, his biographers claimed in retrospect that an oracle further prophesied that the one to untie the knot would become the king of Asia. Alexander is a figure of outstanding celebrity and the episode with the Gordian Knot remains widely known. Literary sources are Alexanders propagandist Arrian Quintus Curtius, Justins epitome of Pompeius Trogus, while sources from antiquity agree that Alexander was confronted with the challenge of the knot, the means by which he solved the problem are disputed. Some classical scholars regard this as more plausible than the popular account, Alexander later went on to conquer Asia as far as the Indus and the Oxus thus, for Callisthenes, fulfilling the prophecy. The knot may have been a religious knot-cipher guarded by Gordian/Midass priests, unlike fable, true myth has few completely arbitrary elements. This myth taken as a whole seems designed to confer legitimacy to change in this central Anatolian kingdom. The ox-cart suggests a longer voyage, rather than a journey, perhaps linking Gordias/Midas with an attested origin-myth in Macedon. The Gordian knot is alluded to in the motto of Ferdinand II of Aragon, Tanto monta and in the yoke representing Isabella in the emblem of the yoke and arrows. Brian Coless has suggested that Donald Wiseman cut the Gordian knot of the problem of identifying King Darius the Mede in the Book of Daniel. W. G. Sebald in The Rings of Saturn recounts the episode of Joseph Conrad who was shot or shot himself in the chest allowing him to Cut the Gordian Knot of, in Sebalds telling, a stormy love affair. Henry Purcell, an English composer, wrote a piece of music entitled The Gordion Knot Untyd and he called for the newborn artists, the anti-Alexanders, to heal the wound and repair the knot, Yes, the rebirth is in the hands of all of us. It is up to us if the West is to bring forth any anti-Alexanders to tie together the Gordian Knot of civilization cut by the sword, for this purpose, we must assume all the risks and labors of freedom
64.
International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker
65.
Link (knot theory)
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In mathematical knot theory, a link is a collection of knots which do not intersect, but which may be linked together. A knot can be described as a link with one component, links and knots are studied in a branch of mathematics called knot theory. Implicit in this definition is that there is a reference link, usually called the unlink. For example, a co-dimension two link in 3-dimensional space is a subspace of 3-dimensional Euclidean space whose connected components are homeomorphic to circles, the simplest nontrivial example of a link with more than one component is called the Hopf link, which consists of two circles linked together once. The circles in the Borromean rings are collectively linked despite the fact that no two of them are directly linked, the Borromean rings thus form a Brunnian link and in fact constitute the simplest such link. The notion of a link can be generalized in a number of ways, frequently the word link is used to describe any submanifold of the sphere S n diffeomorphic to a disjoint union of a finite number of spheres, S j. If M is disconnected, the embedding is called a link, if M is connected, it is called a knot. While links are defined as embeddings of circles, it is interesting and especially technically useful to consider embedded intervals. The type of a tangle is the manifold X, together with an embedding of ∂ X. The condition that the boundary of X lies in R × says that intervals either connect two lines or connect two points on one of the lines, but imposes no conditions on the circles. Tangles include links, braids, and others besides – for example, in this context, a braid is defined as a tangle which is always going down – whose derivative always has a non-zero component in the vertical direction. In particular, it must consist solely of intervals, and not double back on itself, however, a string link is a tangle consisting of only intervals, with the ends of each strand required to lie at. – i. e. connecting the integers, and ending in the order that they began, if this has ℓ components. A string link need not be a braid – it may double back on itself, a braid that is also a string link is called a pure braid, and corresponds with the usual such notion. The key technical value of tangles and string links is that they have algebraic structure, the tensor structure is given by juxtaposition of tangles – putting one tangle to the right of the other. For a fixed ℓ, isotopy classes of ℓ-component string links form a monoid, however, concordance classes of string links do have inverses, where inverse is given by flipping the string link upside down, and thus form a group. Linking number Hyperbolic link Unlink Link group
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Hyperbolic link
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In mathematics, a hyperbolic link is a link in the 3-sphere with complement that has a complete Riemannian metric of constant negative curvature, i. e. has a hyperbolic geometry. A hyperbolic knot is a link with one component. As a consequence of the work of William Thurston, it is known that every knot is one of the following, hyperbolic. As a consequence, hyperbolic knots can be considered plentiful, a similar heuristic applies to hyperbolic links. As a consequence of Thurstons hyperbolic Dehn surgery theorem, performing Dehn surgeries on a hyperbolic link enables one to many more hyperbolic 3-manifolds. Every non-split, prime, alternating link that is not a torus link is hyperbolic by a result of William Menasco. 4₁ knot 5₂ knot 6₁ knot 6₂ knot 6₃ knot 7₄ knot 10161 knot 12n242 knot SnapPea hyperbolic volume Colin Adams The Knot Book, American Mathematical Society, William Menasco Closed incompressible surfaces in alternating knot and link complements, Topology 23, 37–44. William Thurston The geometry and topology of three-manifolds, Princeton lecture notes
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Stevedore knot (mathematics)
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In knot theory, the stevedore knot is one of three prime knots with crossing number six, the others being the 62 knot and the 63 knot. The stevedore knot is listed as the 61 knot in the Alexander–Briggs notation, the mathematical stevedore knot is named after the common stevedore knot, which is often used as a stopper at the end of a rope. The mathematical version of the knot can be obtained from the version by joining together the two loose ends of the rope, forming a knotted loop. The stevedore knot is invertible but not amphichiral, the Alexander polynomial and Conway polynomial are the same as those for the knot 946, but the Jones polynomials for these two knots are different. Because the Alexander polynomial is not monic, the knot is not fibered. The stevedore knot is a knot, and is therefore also a slice knot. The stevedore knot is a knot, with its complement having a volume of approximately 3.16396
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Carrick mat
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The carrick mat is a flat woven decorative knot which can be used as a mat or pad. Its name is based on the mats decorative-type carrick bend with the ends connected together, a larger form, called the prolong knot, is made by expanding the basic carrick mat by extending, twisting, and overlapping its outer bights, then weaving the free ends through them. This process may be repeated to produce an arbitrarily long mat, in its basic form it is the same as a 3-lead, 4-bight Turks head knot. The basic carrick mat, made two passes of rope, also forms the central motif in the logo of the International Guild of Knot Tyers. When tied to form a cylinder around the opening, instead of lying flat. List of knots 8_18, The Knot Atlas
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Whitehead link
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In knot theory, the Whitehead link, named for J. H. C. Whitehead, is one of the most basic links, Whitehead spent much of the 1930s looking for a proof of the Poincaré conjecture. In 1934, the Whitehead link was used as part of his construction of the now-named Whitehead manifold, the link is created with two projections of the unknot, one circular loop and one figure eight-shaped loop intertwined such that they are inseparable and neither loses its form. Excluding the instance where the figure eight thread intersects itself, the Whitehead link has four crossings, because each underhand crossing has a paired upperhand crossing, its linking number is 0. It is not isotopic to the unlink, but it is homotopic to the unlink. In braid theory notation, the link is written σ12 σ22 σ1 −1 σ2 −2 and its Jones polynomial is V = t −32. This polynomial and V are the two factors of the Jones polynomial of the L10a140 link, notably, V is the Jones polynomial for the mirror image of a link having Jones polynomial V. The hyperbolic volume of the complement of the Whitehead link is 4 times Catalans constant, the Whitehead link complement is one of two two-cusped hyperbolic manifolds with the minimum possible volume, the other being the complement of the pretzel link with parameters. Solomons knot Weeks manifold Whitehead double L5a1 knot-theoretic link, The Knot Atlas
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Borromean rings
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In mathematics, the Borromean rings consist of three topological circles which are linked and form a Brunnian link. In other words, no two of the three rings are linked with other as a Hopf link, but nonetheless all three are linked. Although the typical picture of the Borromean rings may lead one to think the link can be formed from geometrically ideal circular rings, freedman and Skora prove that a certain class of links, including the Borromean links, cannot be exactly circular. Alternatively, this can be seen from considering the link diagram, in either case, the third circle must pass through this plane or sphere four times, without lying in it, which is impossible, see. It is, however, true that one can use ellipses and these may be taken to be of arbitrarily small eccentricity, i. e. Apart from indicating which strand crosses over the other, link diagrams use the notation to show two strands crossing, as graph diagrams use to show four edges meeting at a common vertex. The result has three loops, linked together as Borromean rings. In knot theory, the Borromean rings are an example of a Brunnian link, although each pair of rings is unlinked. There are a number of ways of seeing this and this is non-trivial in the fundamental group, and thus the Borromean rings are linked. Another way is that the cohomology of the complement supports a non-trivial Massey product, in arithmetic topology, there is an analogy between knots and prime numbers in which one considers links between primes. The triple of primes are linked modulo 2 but are pairwise unlinked modulo 2, therefore, these primes have been called a proper Borromean triple modulo 2 or mod 2 Borromean primes. The Borromean rings are a link, the complement of the Borromean rings in the 3-sphere admits a complete hyperbolic metric of finite volume. The canonical polyhedral decomposition of the complement consists of two regular ideal octahedra, the volume is 16Л =7. 32772… where Л is the Lobachevsky function. If one cuts the Borromean rings, one obtains one iteration of the braid, conversely, if one ties together the ends of a standard braid. Just as removing one Borromean ring unlinks the remaining two, removing one strand of the standard braid unbraids the other two, they are the basic Brunnian link and Brunnian braid, respectively. In the standard link diagram, the Borromean rings are ordered non-transitively, in a cyclic order. Using the colors above, these are red over green, green over blue, blue over red – and thus removing any one ring. Similarly, in the standard braid, each strand is above one of the others, the name Borromean rings comes from their use in the coat of arms of the aristocratic Borromeo family in Northern Italy
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L10a140 link
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In the mathematical theory of knots, L10a140 is the name in the Thistlewaite link table of a link of three loops, which has ten crossings between the loops when presented in its simplest visual form. In other words, no two loops are linked with each other, but all three are collectively interlinked, so removing any loop frees the other two. According to work by Slavik V. Jablan, the L10a140 link can be seen as the second in an series of Brunnian links beginning with the Borromean rings. David Swart, and independently Rick Mabry and Laura McCormick, discovered alternative 12-crossing visual representations of the L10a140 link, in these depictions, the link no longer has strictly alternating crossings, but there is greater superficial symmetry. So the leftmost image below shows a 12-crossing link with six-fold rotational symmetry, the center image shows a similar-looking depiction of the L10a140 link. Similarly, the rightmost image shows a depiction of the L10a140 link with superficial fourfold symmetry and it Is What It Is, Flickr. com
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Satellite knot
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In the mathematical theory of knots, a satellite knot is a knot that contains an incompressible, non boundary-parallel torus in its complement. Every knot is either hyperbolic, a torus, or a satellite knot, the class of satellite knots include composite knots, cable knots and Whitehead doubles. A satellite link is one that orbits a companion knot K in the sense that it lies inside a regular neighborhood of the companion, a satellite knot K can be picturesquely described as follows, start by taking a nontrivial knot K ′ lying inside an unknotted solid torus V. Here nontrivial means that the knot K ′ is not allowed to sit inside of a 3-ball in V and K ′ is not allowed to be isotopic to the central curve of the solid torus. Then tie up the solid torus into a nontrivial knot and this means there is a non-trivial embedding f, V → S3 and K = f. Since V is a solid torus, S3 ∖ V is a tubular neighbourhood of an unknot J. The 2-component link K ′ ∪ J together with the embedding f is called the pattern associated to the satellite operation. A convention, people demand that the embedding f, V → S3 is untwisted in the sense that f must send the standard longitude of V to the standard longitude of f. Said another way, given two disjoint curves c 1, c 2 ⊂ V, f must preserve their linking numbers i. e. l k = l k, when K ′ ⊂ ∂ V is a torus knot, then K is called a cable knot. Examples 3 and 4 are cable knots, if K ′ is a non-trivial knot in S3 and if a compressing disc for V intersects K ′ in precisely one point, then K is called a connect-sum. Another way to say this is that the pattern K ′ ∪ J is the connect-sum of a non-trivial knot K ′ with a Hopf link, if the link K ′ ∪ J is the Whitehead link, K is called a Whitehead double. If f is untwisted, K is called an untwisted Whitehead double, Example 1, The connect-sum of a figure-8 knot and trefoil. Example 2, Untwisted Whitehead double of a figure-8, Example 3, Cable of a connect-sum. Examples 5 and 6 are variants on the same construction and they both have two non-parallel, non-boundary-parallel incompressible tori in their complements, splitting the complement into the union of three manifolds. In Example 5 those manifolds are, the Borromean rings complement, trefoil complement, in Example 6 the figure-8 complement is replaced by another trefoil complement. Shortly after, he realized he could give a new proof of his theorem by an analysis of the incompressible tori present in the complement of a connect-sum. This led him to study general incompressible tori in knot complements in his epic work Knoten und Vollringe, schuberts demonstration that incompressible tori play a major role in knot theory was one several early insights leading to the unification of 3-manifold theory and knot theory. It attracted Waldhausens attention, who later used incompressible surfaces to show that a class of 3-manifolds are homeomorphic if