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
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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
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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
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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
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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
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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