1.
Braid theory
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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

2.
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

3.
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

4.
Seifert surface
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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

5.
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

6.
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

7.
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

8.
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

9.
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

10.
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

11.
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

12.
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

13.
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

14.
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

15.
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

16.
Chiral 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

17.
Trefoil knot
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In topology, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot. The trefoil can be obtained by joining together the two ends of a common overhand knot, resulting in a knotted loop. As the simplest knot, the trefoil is fundamental to the study of knot theory. The trefoil knot is named after the three-leaf clover plant, specifically, any curve isotopic to a trefoil knot is also considered to be a trefoil. In addition, the image of a trefoil knot is also considered to be a trefoil. In topology and knot theory, the trefoil is usually defined using a knot diagram instead of a parametric equation. In algebraic geometry, the trefoil can also be obtained as the intersection in C2 of the unit 3-sphere S3 with the plane curve of zeroes of the complex polynomial z2 + w3. If one end of a tape or belt is turned over three times and then pasted to the other, the forms a trefoil knot. The trefoil knot is chiral, in the sense that a knot can be distinguished from its own mirror image. The two resulting variants are known as the trefoil and the right-handed trefoil. It is not possible to deform a left-handed trefoil continuously into a right-handed trefoil, though the trefoil knot is chiral, it is also invertible, meaning that there is no distinction between a counterclockwise-oriented trefoil and a clockwise-oriented trefoil. That is, the chirality of a trefoil depends only on the over and under crossings, the trefoil knot is nontrivial, meaning that it is not possible to untie a trefoil knot in three dimensions without cutting it. From a mathematical point of view, this means that a knot is not isotopic to the unknot. In particular, there is no sequence of Reidemeister moves that will untie a trefoil, proving this requires the construction of a knot invariant that distinguishes the trefoil from the unknot. The simplest such invariant is tricolorability, the trefoil is tricolorable, in addition, virtually every major knot polynomial distinguishes the trefoil from an unknot, as do most other strong knot invariants. In knot theory, the trefoil is the first nontrivial knot and it is a prime knot, and is listed as 31 in the Alexander-Briggs notation. The Dowker notation for the trefoil is 462, the trefoil can be described as the -torus knot. It is also the knot obtained by closing the braid σ13, the trefoil is an alternating knot

18.
Cinquefoil knot
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In knot theory, the cinquefoil knot, also known as Solomons seal knot or the pentafoil knot, is one of two knots with crossing number five, the other being the three-twist knot. It is listed as the 51 knot in the Alexander-Briggs notation, the cinquefoil is the closed version of the double overhand knot. The cinquefoil is a prime knot and its writhe is 5, and it is invertible but not amphichiral. These are the same as the Alexander, Conway, and Jones polynomials of the knot 10132, however, the Kauffman polynomial can be used to distinguish between these two knots. The name “cinquefoil” comes from the flowers of plants in the genus Potentilla. Pentagram Trefoil knot 7₁ knot Skein relation A Pentafoil Knot at the Wayback Machine

19.
Invertible knot
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In mathematics, especially in the area of topology known as knot theory, an invertible knot is a knot that can be continuously deformed to itself, but with its orientation reversed. A non-invertible knot is any knot which does not have this property, the invertibility of a knot is a knot invariant. An invertible link is the equivalent of an invertible knot. It has long known that most of the simple knots, such as the trefoil knot. It is now known almost all knots are non-invertible, all knots with crossing number of 7 or less are known to be invertible. No general method is known that can distinguish if a knot is invertible. The problem can be translated into algebraic terms, but unfortunately there is no algorithm to solve this algebraic problem. If a knot is invertible and amphichiral, it is fully amphichiral, the simplest knot with this property is the figure eight knot. A chiral knot that is invertible is classified as a reversible knot, a more abstract way to define an invertible knot is to say there is an orientation-preserving homeomorphism of the 3-sphere which takes the knot to itself but reverses the orientation along the knot. All knots with tunnel number one, such as the trefoil knot, the simplest example of a non-invertible knot is the knot 817 or.2.2. The pretzel knot 7,5,3 is non-invertible, as are all pretzel knots of the form, where p, q, and r are distinct integers, chiral knot Jablan, Slavik & Sazdanovic, Radmila. Basic graph theory, Non-invertible knot and links, LinKnot

20.
Alexander polynomial
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In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered this, the first knot polynomial, soon after Conways reworking of the Alexander polynomial, it was realized that a similar skein relation was exhibited in Alexanders paper on his polynomial. Let K be a knot in the 3-sphere, let X be the infinite cyclic cover of the knot complement of K. This covering can be obtained by cutting the knot complement along a Seifert surface of K, there is a covering transformation t acting on X. Consider the first homology of X, denoted H1, the transformation t acts on the homology and so we can consider H1 a module over Z. This is called the Alexander invariant or Alexander module, the module is finitely presentable, a presentation matrix for this module is called the Alexander matrix. If r > s, set the equal to 0. If the Alexander ideal is principal, take a generator, this is called an Alexander polynomial of the knot, since this is only unique up to multiplication by the Laurent monomial ± t n, one often fixes a particular unique form. Alexanders choice of normalization is to make the polynomial have a constant term. Alexander proved that the Alexander ideal is nonzero and always principal, thus an Alexander polynomial always exists, and is clearly a knot invariant, denoted Δ K. The Alexander polynomial for the knot configured by only one string is a polynomial of t2, namely, it can not distinguish between the knot and one for its mirror image. The following procedure for computing the Alexander polynomial was given by J. W. Alexander in his paper, take an oriented diagram of the knot with n crossings, there are n +2 regions of the knot diagram. To work out the Alexander polynomial, first one must create a matrix of size. The n rows correspond to the n crossings, and the n +2 columns to the regions, the values for the matrix entries are either 0,1, −1, t, −t. Consider the entry corresponding to a region and crossing. If the region is not adjacent to the crossing, the entry is 0, if the region is adjacent to the crossing, the entry depends on its location. The following table gives the entry, determined by the location of the region at the crossing from the perspective of the incoming undercrossing line, depending on the columns removed, the answer will differ by multiplication by ± t n. To resolve this ambiguity, divide out the largest possible power of t and multiply by −1 if necessary, the Alexander polynomial can also be computed from the Seifert matrix

21.
Jones polynomial
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In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. Specifically, it is an invariant of a knot or link which assigns to each oriented knot or link a Laurent polynomial in the variable t 1 /2 with integer coefficients. Suppose we have an oriented link L, given as a knot diagram and we will define the Jones polynomial, V, using Kauffmans bracket polynomial, which we denote by ⟨ ⟩. Note that here the bracket polynomial is a Laurent polynomial in the variable A with integer coefficients, first, we define the auxiliary polynomial X = − w ⟨ L ⟩, where w denotes the writhe of L in its given diagram. The writhe of a diagram is the number of positive crossings minus the number of negative crossings, the writhe is not a knot invariant. X is a knot invariant since it is invariant under changes of the diagram of L by the three Reidemeister moves, invariance under type II and III Reidemeister moves follows from invariance of the bracket under those moves. The bracket polynomial is known to change by multiplication by − A ±3 under a type I Reidemeister move, the definition of the X polynomial given above is designed to nullify this change, since the writhe changes appropriately by +1 or -1 under type I moves. Now make the substitution A = t −1 /4 in X to get the Jones polynomial V and this results in a Laurent polynomial with integer coefficients in the variable t 1 /2. This construction of the Jones Polynomial for tangles is a generalization of the Kauffman bracket of a link. The construction was developed by Professor Vladimir G. Turaev and published in 1990 in the Journal of Mathematics and Science. Let k be an integer and S k denote the set of all isotopic types of tangle diagrams, with 2 k ends, having no crossing points. Jones original formulation of his polynomial came from his study of operator algebras, a theorem of Alexanders states that it is the trace closure of a braid, say with n strands. Now define a representation ρ of the group on n strands, Bn. The standard braid generator σ i is sent to A ⋅ e i + A −1 ⋅1 and it can be checked easily that this defines a representation. Take the braid word σ obtained previously from L and compute δ n −1 t r ρ where tr is the Markov trace and this gives ⟨ L ⟩, where ⟨ ⟩ is the bracket polynomial. This can be seen by considering, as Kauffman did, the Temperley–Lieb algebra as a particular diagram algebra, an advantage of this approach is that one can pick similar representations into other algebras, such as the R-matrix representations, leading to generalized Jones invariants. Thus, a knot, a knot equivalent to its mirror image, has palindromic entries in its Jones polynomial. See the article on skein relation for an example of a computation using these relations, another remarkable property of this invariant states that the Jones Polynomial of an alternating link is an alternating polynomial

22.
Heptagram
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A heptagram, septagram, or septogram is a seven-point star drawn with seven straight strokes. The name heptagram combines a numeral prefix, hepta-, with the Greek suffix -gram, the -gram suffix derives from γραμμῆς meaning a line. In general, a heptagram is any self-intersecting heptagon, there are two regular heptagrams, labeled as and, with the second number representing the vertex interval step from a regular heptagon. This is the smallest star polygon that can be drawn in two forms, as irreducible fractions, the two heptagrams are sometimes called the heptagram and the great heptagram. The previous one, the hexagram, is a compound of two triangles. The smallest star polygon is the pentagram, the next one is the octagram, followed by the regular enneagram, which also has two forms, and, as well as one compound of three triangles. The heptagram was used in Christianity to symbolize the seven days of creation, the heptagram is a symbol of perfection in many Christian sects. The heptagram is used in the symbol for Babalon in Thelema, the heptagram is known among neopagans as the Elven Star or Fairy Star. It is treated as a symbol in various modern pagan. Blue Star Wicca also uses the symbol, where it is referred to as a septegram, the second heptagram is a symbol of magical power in some pagan spiritualities. The heptagram is used by members of the otherkin subculture as an identifier. In alchemy, a star can refer to the seven planets which were known to ancient alchemists. The seven-pointed star is incorporated into the flags of the bands of the Cherokee Nation. The Bennington flag, a historical American Flag, has thirteen seven-pointed stars along with the numerals 76 in the canton, the Flag of Jordan contains a seven-pointed star. The Flag of Australia employs five heptagrams and one pentagram to depict the Southern Cross constellation, some old versions of the coat of arms of Georgia including the Georgian Soviet Socialist Republic used the heptagram as an element. A seven-pointed star is used as the badge in many sheriffs departments, the seven-pointed star is used as the logo for the international Danish shipping company A. P. Moller–Maersk Group, sometimes known simply as Maersk. In George R. R. Martins novel series A Song of Ice and Fire, Star polygon Stellated polygons Two-dimensional regular polytopes Bibliography Grünbaum, B. and G. C. Shephard, Tilings and Patterns, New York, W. H. Freeman & Co, polyhedra with Hollow Faces, Proc of NATO-ASI Conference on Polytopes

23.
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

24.
Link (knot theory)
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In mathematical knot theory, a link is a collection of knots which do not intersect, but which may be linked together. A knot can be described as a link with one component, links and knots are studied in a branch of mathematics called knot theory. Implicit in this definition is that there is a reference link, usually called the unlink. For example, a co-dimension two link in 3-dimensional space is a subspace of 3-dimensional Euclidean space whose connected components are homeomorphic to circles, the simplest nontrivial example of a link with more than one component is called the Hopf link, which consists of two circles linked together once. The circles in the Borromean rings are collectively linked despite the fact that no two of them are directly linked, the Borromean rings thus form a Brunnian link and in fact constitute the simplest such link. The notion of a link can be generalized in a number of ways, frequently the word link is used to describe any submanifold of the sphere S n diffeomorphic to a disjoint union of a finite number of spheres, S j. If M is disconnected, the embedding is called a link, if M is connected, it is called a knot. While links are defined as embeddings of circles, it is interesting and especially technically useful to consider embedded intervals. The type of a tangle is the manifold X, together with an embedding of ∂ X. The condition that the boundary of X lies in R × says that intervals either connect two lines or connect two points on one of the lines, but imposes no conditions on the circles. Tangles include links, braids, and others besides – for example, in this context, a braid is defined as a tangle which is always going down – whose derivative always has a non-zero component in the vertical direction. In particular, it must consist solely of intervals, and not double back on itself, however, a string link is a tangle consisting of only intervals, with the ends of each strand required to lie at. – i. e. connecting the integers, and ending in the order that they began, if this has ℓ components. A string link need not be a braid – it may double back on itself, a braid that is also a string link is called a pure braid, and corresponds with the usual such notion. The key technical value of tangles and string links is that they have algebraic structure, the tensor structure is given by juxtaposition of tangles – putting one tangle to the right of the other. For a fixed ℓ, isotopy classes of ℓ-component string links form a monoid, however, concordance classes of string links do have inverses, where inverse is given by flipping the string link upside down, and thus form a group. Linking number Hyperbolic link Unlink Link group

25.
Hyperbolic link
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In mathematics, a hyperbolic link is a link in the 3-sphere with complement that has a complete Riemannian metric of constant negative curvature, i. e. has a hyperbolic geometry. A hyperbolic knot is a link with one component. As a consequence of the work of William Thurston, it is known that every knot is one of the following, hyperbolic. As a consequence, hyperbolic knots can be considered plentiful, a similar heuristic applies to hyperbolic links. As a consequence of Thurstons hyperbolic Dehn surgery theorem, performing Dehn surgeries on a hyperbolic link enables one to many more hyperbolic 3-manifolds. Every non-split, prime, alternating link that is not a torus link is hyperbolic by a result of William Menasco. 4₁ knot 5₂ knot 6₁ knot 6₂ knot 6₃ knot 7₄ knot 10161 knot 12n242 knot SnapPea hyperbolic volume Colin Adams The Knot Book, American Mathematical Society, William Menasco Closed incompressible surfaces in alternating knot and link complements, Topology 23, 37–44. William Thurston The geometry and topology of three-manifolds, Princeton lecture notes

26.
Figure-eight knot (mathematics)
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In knot theory, a figure-eight knot is the unique knot with a crossing number of four. This is the smallest possible crossing number except for the unknot, the figure-eight knot is a prime knot. The name is given because tying a normal figure-eight knot in a rope and then joining the ends together, in the most natural way, gives a model of the mathematical knot. A simple parametric representation of the knot is as the set of all points where x = cos y = sin z = sin for t varying over the real numbers. The figure-eight knot is prime, alternating, rational with a value of 5/2. The figure-eight knot is also a fibered knot and this follows from other, less simple representations of the knot, It is a homogeneous closed braid, and a theorem of John Stallings shows that any closed homogeneous braid is fibered. It is the link at of a critical point of a real-polynomial map F, R4→R2. Bernard Perron found the first such F for this knot, namely, F = G, the figure-eight knot has played an important role historically in the theory of 3-manifolds. Sometime in the mid-to-late 1970s, William Thurston showed that the figure-eight was hyperbolic and this construction, new at the time, led him to many powerful results and methods. For example, he was able to show all but ten Dehn surgeries on the figure-eight knot resulted in non-Haken, non-Seifert-fibered irreducible 3-manifolds. Many more have been discovered by generalizing Thurstons construction to other knots, the figure-eight knot is also the hyperbolic knot whose complement has the smallest possible volume,2.02988. According to the work of Chun Cao and Robert Meyerhoff, from this perspective, the figure-eight knot can be considered the simplest hyperbolic knot. The figure eight knot complement is a double-cover of the Gieseking manifold, however, it is not currently known whether the figure-eight knot is the only one that achieves the bound of 10. A well-known conjecture is that the bound is 6, the symmetry between q and q −1 in the Jones polynomial reflects the fact that the figure-eight knot is achiral. Ian Agol, Bounds on exceptional Dehn filling, Geometry & Topology 4, mR1799796 Chun Cao and Robert Meyerhoff, The orientable cusped hyperbolic 3-manifolds of minimum volume, Inventiones Mathematicae,146, no. MR1869847 Marc Lackenby, Word hyperbolic Dehn surgery, Inventiones Mathematicae 140, no

27.
Stevedore knot (mathematics)
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In knot theory, the stevedore knot is one of three prime knots with crossing number six, the others being the 62 knot and the 63 knot. The stevedore knot is listed as the 61 knot in the Alexander–Briggs notation, the mathematical stevedore knot is named after the common stevedore knot, which is often used as a stopper at the end of a rope. The mathematical version of the knot can be obtained from the version by joining together the two loose ends of the rope, forming a knotted loop. The stevedore knot is invertible but not amphichiral, the Alexander polynomial and Conway polynomial are the same as those for the knot 946, but the Jones polynomials for these two knots are different. Because the Alexander polynomial is not monic, the knot is not fibered. The stevedore knot is a knot, and is therefore also a slice knot. The stevedore knot is a knot, with its complement having a volume of approximately 3.16396

28.
Carrick mat
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The carrick mat is a flat woven decorative knot which can be used as a mat or pad. Its name is based on the mats decorative-type carrick bend with the ends connected together, a larger form, called the prolong knot, is made by expanding the basic carrick mat by extending, twisting, and overlapping its outer bights, then weaving the free ends through them. This process may be repeated to produce an arbitrarily long mat, in its basic form it is the same as a 3-lead, 4-bight Turks head knot. The basic carrick mat, made two passes of rope, also forms the central motif in the logo of the International Guild of Knot Tyers. When tied to form a cylinder around the opening, instead of lying flat. List of knots 8_18, The Knot Atlas

29.
Whitehead link
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In knot theory, the Whitehead link, named for J. H. C. Whitehead, is one of the most basic links, Whitehead spent much of the 1930s looking for a proof of the Poincaré conjecture. In 1934, the Whitehead link was used as part of his construction of the now-named Whitehead manifold, the link is created with two projections of the unknot, one circular loop and one figure eight-shaped loop intertwined such that they are inseparable and neither loses its form. Excluding the instance where the figure eight thread intersects itself, the Whitehead link has four crossings, because each underhand crossing has a paired upperhand crossing, its linking number is 0. It is not isotopic to the unlink, but it is homotopic to the unlink. In braid theory notation, the link is written σ12 σ22 σ1 −1 σ2 −2 and its Jones polynomial is V = t −32. This polynomial and V are the two factors of the Jones polynomial of the L10a140 link, notably, V is the Jones polynomial for the mirror image of a link having Jones polynomial V. The hyperbolic volume of the complement of the Whitehead link is 4 times Catalans constant, the Whitehead link complement is one of two two-cusped hyperbolic manifolds with the minimum possible volume, the other being the complement of the pretzel link with parameters. Solomons knot Weeks manifold Whitehead double L5a1 knot-theoretic link, The Knot Atlas

30.
Borromean rings
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In mathematics, the Borromean rings consist of three topological circles which are linked and form a Brunnian link. In other words, no two of the three rings are linked with other as a Hopf link, but nonetheless all three are linked. Although the typical picture of the Borromean rings may lead one to think the link can be formed from geometrically ideal circular rings, freedman and Skora prove that a certain class of links, including the Borromean links, cannot be exactly circular. Alternatively, this can be seen from considering the link diagram, in either case, the third circle must pass through this plane or sphere four times, without lying in it, which is impossible, see. It is, however, true that one can use ellipses and these may be taken to be of arbitrarily small eccentricity, i. e. Apart from indicating which strand crosses over the other, link diagrams use the notation to show two strands crossing, as graph diagrams use to show four edges meeting at a common vertex. The result has three loops, linked together as Borromean rings. In knot theory, the Borromean rings are an example of a Brunnian link, although each pair of rings is unlinked. There are a number of ways of seeing this and this is non-trivial in the fundamental group, and thus the Borromean rings are linked. Another way is that the cohomology of the complement supports a non-trivial Massey product, in arithmetic topology, there is an analogy between knots and prime numbers in which one considers links between primes. The triple of primes are linked modulo 2 but are pairwise unlinked modulo 2, therefore, these primes have been called a proper Borromean triple modulo 2 or mod 2 Borromean primes. The Borromean rings are a link, the complement of the Borromean rings in the 3-sphere admits a complete hyperbolic metric of finite volume. The canonical polyhedral decomposition of the complement consists of two regular ideal octahedra, the volume is 16Л =7. 32772… where Л is the Lobachevsky function. If one cuts the Borromean rings, one obtains one iteration of the braid, conversely, if one ties together the ends of a standard braid. Just as removing one Borromean ring unlinks the remaining two, removing one strand of the standard braid unbraids the other two, they are the basic Brunnian link and Brunnian braid, respectively. In the standard link diagram, the Borromean rings are ordered non-transitively, in a cyclic order. Using the colors above, these are red over green, green over blue, blue over red – and thus removing any one ring. Similarly, in the standard braid, each strand is above one of the others, the name Borromean rings comes from their use in the coat of arms of the aristocratic Borromeo family in Northern Italy

31.
L10a140 link
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In the mathematical theory of knots, L10a140 is the name in the Thistlewaite link table of a link of three loops, which has ten crossings between the loops when presented in its simplest visual form. In other words, no two loops are linked with each other, but all three are collectively interlinked, so removing any loop frees the other two. According to work by Slavik V. Jablan, the L10a140 link can be seen as the second in an series of Brunnian links beginning with the Borromean rings. David Swart, and independently Rick Mabry and Laura McCormick, discovered alternative 12-crossing visual representations of the L10a140 link, in these depictions, the link no longer has strictly alternating crossings, but there is greater superficial symmetry. So the leftmost image below shows a 12-crossing link with six-fold rotational symmetry, the center image shows a similar-looking depiction of the L10a140 link. Similarly, the rightmost image shows a depiction of the L10a140 link with superficial fourfold symmetry and it Is What It Is, Flickr. com

32.
Satellite knot
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In the mathematical theory of knots, a satellite knot is a knot that contains an incompressible, non boundary-parallel torus in its complement. Every knot is either hyperbolic, a torus, or a satellite knot, the class of satellite knots include composite knots, cable knots and Whitehead doubles. A satellite link is one that orbits a companion knot K in the sense that it lies inside a regular neighborhood of the companion, a satellite knot K can be picturesquely described as follows, start by taking a nontrivial knot K ′ lying inside an unknotted solid torus V. Here nontrivial means that the knot K ′ is not allowed to sit inside of a 3-ball in V and K ′ is not allowed to be isotopic to the central curve of the solid torus. Then tie up the solid torus into a nontrivial knot and this means there is a non-trivial embedding f, V → S3 and K = f. Since V is a solid torus, S3 ∖ V is a tubular neighbourhood of an unknot J. The 2-component link K ′ ∪ J together with the embedding f is called the pattern associated to the satellite operation. A convention, people demand that the embedding f, V → S3 is untwisted in the sense that f must send the standard longitude of V to the standard longitude of f. Said another way, given two disjoint curves c 1, c 2 ⊂ V, f must preserve their linking numbers i. e. l k = l k, when K ′ ⊂ ∂ V is a torus knot, then K is called a cable knot. Examples 3 and 4 are cable knots, if K ′ is a non-trivial knot in S3 and if a compressing disc for V intersects K ′ in precisely one point, then K is called a connect-sum. Another way to say this is that the pattern K ′ ∪ J is the connect-sum of a non-trivial knot K ′ with a Hopf link, if the link K ′ ∪ J is the Whitehead link, K is called a Whitehead double. If f is untwisted, K is called an untwisted Whitehead double, Example 1, The connect-sum of a figure-8 knot and trefoil. Example 2, Untwisted Whitehead double of a figure-8, Example 3, Cable of a connect-sum. Examples 5 and 6 are variants on the same construction and they both have two non-parallel, non-boundary-parallel incompressible tori in their complements, splitting the complement into the union of three manifolds. In Example 5 those manifolds are, the Borromean rings complement, trefoil complement, in Example 6 the figure-8 complement is replaced by another trefoil complement. Shortly after, he realized he could give a new proof of his theorem by an analysis of the incompressible tori present in the complement of a connect-sum. This led him to study general incompressible tori in knot complements in his epic work Knoten und Vollringe, schuberts demonstration that incompressible tori play a major role in knot theory was one several early insights leading to the unification of 3-manifold theory and knot theory. It attracted Waldhausens attention, who later used incompressible surfaces to show that a class of 3-manifolds are homeomorphic if

33.
Granny knot (mathematics)
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In knot theory, the granny knot is a composite knot obtained by taking the connected sum of two identical trefoil knots. It is closely related to the knot, which can also be described as a connected sum of two trefoils. Because the trefoil knot is the simplest nontrivial knot, the granny knot, the granny knot is the mathematical version of the common granny knot. The granny knot can be constructed from two trefoil knots, which must either be both left-handed or both right-handed. Each of the two knots is cut, and then the loose ends are joined together pairwise, the resulting connected sum is the granny knot. It is important that the trefoil knots be identical to each another. If mirror-image trefoil knots are used instead, the result is a square knot, the crossing number of a granny knot is six, which is the smallest possible crossing number for a composite knot. Unlike the square knot, the knot is not a ribbon knot or a slice knot. The Alexander polynomial of the knot is Δ =2. Similarly, the Conway polynomial of a knot is ∇ =2. These two polynomials are the same as those for the square knot. However, the Jones polynomial for the knot is V =2 = q −2 +2 q −4 −2 q −5 + q −6 −2 q −7 + q −8. This is the square of the Jones polynomial for the trefoil knot. The knot group of the knot is given by the presentation ⟨ x, y, z ∣ x y x = y x y, x z x = z x z ⟩. This is isomorphic to the group of the square knot

34.
Square knot (mathematics)
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In knot theory, the square knot is a composite knot obtained by taking the connected sum of a trefoil knot with its reflection. It is closely related to the knot, which is also a connected sum of two trefoils. Because the trefoil knot is the simplest nontrivial knot, the square knot, the square knot is the mathematical version of the common reef knot. The square knot can be constructed from two trefoil knots, one of which must be left-handed and the other right-handed, each of the two knots is cut, and then the loose ends are joined together pairwise. The resulting connected sum is the square knot and it is important that the original trefoil knots be mirror images of one another. If two identical trefoil knots are used instead, the result is a granny knot, the square knot is amphichiral, meaning that it is indistinguishable from its own mirror image. The crossing number of a knot is six, which is the smallest possible crossing number for a composite knot. The Alexander polynomial of the knot is Δ =2. Similarly, the Alexander–Conway polynomial of a knot is ∇ =2. These two polynomials are the same as those for the granny knot, however, the Jones polynomial for the square knot is V = = − q 3 + q 2 − q +3 − q −1 + q −2 − q −3. This is the product of the Jones polynomials for the right-handed and left-handed trefoil knots, the knot group of the square knot is given by the presentation ⟨ x, y, z ∣ x y x = y x y, x z x = z x z ⟩. This is isomorphic to the group of the granny knot. Unlike the granny knot, the knot is a ribbon knot

35.
Connected sum
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In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to two given manifolds together near a chosen point on each. This construction plays a key role in the classification of closed surfaces, more generally, one can also join manifolds together along identical submanifolds, this generalization is often called the fiber sum. There is also a related notion of a connected sum on knots. A connected sum of two m-dimensional manifolds is a formed by deleting a ball inside each manifold and gluing together the resulting boundary spheres. If both manifolds are oriented, there is a connected sum defined by having the gluing map reverse orientation. Although the construction uses the choice of the balls, the result is unique up to homeomorphism, one can also make this operation work in the smooth category, and then the result is unique up to diffeomorphism. There are subtle problems in the case, not every diffeomorphism between the boundaries of the spheres gives the same composite manifold, even if the orientations are chosen correctly. For example, Milnor showed that two 7-cells can be glued along their boundary so that the result is an exotic sphere homeomorphic, however, there is a canonical way to choose the gluing of M1 and M2 which gives a unique well defined connected sum. Choose embeddings i 1, D n → M1 and i 2, D n → M2 so that i 1 preserves orientation and i 2 reverses orientation. Now obtain M1 # M2 from the disjoint sum ⊔ by identifying i 1 with i 2 for each unit vector u ∈ S n −1, choose the orientation for M1 # M2 which is compatible with M1 and M2. The fact that construction is well-defined depends crucially on the disc theorem. For further details, see The operation of connected sum is denoted by #, the operation of connected sum has the sphere S m as an identity, that is, M # S m is homeomorphic to M. Let M1 and M2 be two smooth, oriented manifolds of dimension and V a smooth, closed, oriented manifold. Suppose furthermore that there exists an isomorphism of normal bundles ψ, N M1 V → N M2 V that reverses the orientation on each fiber. The connected sum of M1 and M2 along V is then the space ⋃ N1 ∖ V = N2 ∖ V obtained by gluing the deleted neighborhoods together by the orientation-preserving diffeomorphism, the sum is often denoted #. Its diffeomorphism type depends on the choice of the two embeddings of V and on the choice of ψ, for this reason, the connected sum along V is often called the fiber sum. The special case of V a point recovers the connected sum of the preceding section, another important special case occurs when the dimension of V is two less than that of the M i

36.
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

37.
Unlink
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In the mathematical field of knot theory, the unlink is a link that is equivalent to finitely many disjoint circles in the plane. An n-component link L ⊂ S3 is an if and only if there exists n disjointly embedded discs Di ⊂ S3 such that L = ∪i∂Di. A link with one component is an if and only if it is the unknot. The link group of an n-component unlink is the group on n generators. The Hopf link is an example of a link with two components that is not an unlink. The Borromean rings form a link with three components that is not an unlink, however, any two of the rings considered on their own do form a two-component unlink. Kanenobu has shown that for all n >1 there exists a link of n components such that any proper sublink is an unlink. The Whitehead link and Borromean rings are such examples for n =2,3

38.
Hopf link
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In mathematical knot theory, the Hopf link is the simplest nontrivial link with more than one component. It consists of two circles linked together once, and is named after Heinz Hopf. A concrete model consists of two circles in perpendicular planes, each passing through the center of the other. This model minimizes the ropelength of the link and until 2002 the Hopf link was the only link whose ropelength was known, the convex hull of these two circles forms a shape called an oloid. Depending on the orientations of the two components the linking number of the Hopf link is ±1. The Hopf link is a link with the braid word σ12. The knot complement of the Hopf link is R × S1 × S1 and this space has a locally Euclidean geometry, so the Hopf link is not a hyperbolic link. The knot group of the Hopf link is Z2, distinguishing it from a pair of loops which has the free group on two generators as its group. This is easily seen from the fact that the link can only take on two colors that leads it to fail the second part of the definition of tricoloribility, at each crossing, it will take a maximum of 2 colors. Thus, if it satisfies the rule of having more than 1 color, if it satisfies the rule of having 1 or 3 colors at each crossing, it will fail the rule of having more than 1 color. The Hopf fibration is a function from the 3-sphere into the more familiar 2-sphere. Thus, these images decompose the 3-sphere into a family of circles. This was Hopfs motivation for studying the Hopf link, because each two fibers are linked, the Hopf fibration is a nontrivial fibration and this example began the study of homotopy groups of spheres. The Hopf link is also present in some proteins and it is consists of two covalent loops, formed by pieces of protein backbone, closed with disulfide bonds. The Hopf link topology is highly conserved in proteins and ads to their stability, the Hopf link is named after topologist Heinz Hopf, who considered it in 1931 as part of his research on the Hopf fibration. However, in mathematics, it was known to Carl Friedrich Gauss before the work of Hopf and it has also long been used outside mathematics, for instance as the crest of Buzan-ha, a Japanese Buddhist sect founded in the 16th century. Catenane, a molecule with two linked loops Solomons knot, two loops which are doubly linked Weisstein, Eric W. Hopf Link, linkProt - the database of known protein links

39.
Solomon's knot
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Solomons knot is the most common name for a traditional decorative motif used since ancient times, and found in many cultures. Despite the name, it is classified as a link, and is not a true knot according to the definitions of mathematical knot theory, the Solomons knot consists of two closed loops, which are doubly interlinked in an interlaced manner. If laid flat, the Solomons knot is seen to have four crossings where the two loops interweave under and over each other and this contrasts with two crossings in the simpler Hopf link. In most artistic representations, the parts of the loops that alternately cross over and under each other become the sides of a central square, while four loopings extend outward in four directions. The four extending loopings may have oval, square, or triangular endings, or may terminate with free-form shapes such as leaves, lobes, blades, the Solomons knot often occurs in ancient Roman mosaics, usually represented as two interlaced ovals. Tzippori National Park, Israel, has Solomons Knots in stone mosaics at the site of an ancient synagogue, across the Middle East, historical Islamic sites show Solomons knot as part of Muslim tradition. It appears over the doorway of a twentieth century CE mosque/madrasa in Cairo. Two versions of Solomons knot are included in the recently excavated Yattir Mosaic in Jordan, to the east, it is woven into an antique Central Asian prayer rug. To the west, Solomons knot appeared in Moorish Spain, the British Museum, London, England has a fourteenth-century CE Egyptian Quran with a Solomons Knot as its frontispiece. Home of Peace Mausoleum, a Jewish Cemetery, Los Angeles, California, USA has multiple images of Solomons knot in stone, saint Sophias Greek Orthodox Cathedral, Byzantine District of Los Angeles, California, USA has an olive wood Epitaphios with Solomons knots carved at each corner. The Epitaphios is used in the Greek Easter services, powell Library University of California at Los Angeles, USA has ceiling beams in the Main Reading Room covered with Solomons Knots. Built in 1926 CE, the room also features a central Dome of Wisdom bordered by Solomons knots. In Latin, this configuration was sometimes known as sigillum Salomonis and it was associated with the Biblical monarch Solomon because of his reputation for wisdom and knowledge. This phrase is rendered into English as Solomons knot, since seal of Solomon has other conflicting meanings. Imbolo describes the design on the textiles of the Kuba people of Congo. Nodo di Salomone is the Italian term for Solomons Knot, and is used to name the Solomons Knot mosaic found at the ruins of a synagogue at Ostia, the ancient seaport for Rome. Since the knot has been used across a number of cultures and historical eras, because there is no visible beginning or ending, it may represent immortality and eternity—as does the more complicated Buddhist Endless Knot. Because the knot seems to be two entwined figures, it is interpreted as a Lovers Knot, although that name may indicate another knot

40.
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

41.
2-bridge knot
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Equivalently, these are the knots with bridge number 2, the smallest possible bridge number for a nontrivial knot. Other names for 2-bridge knots are rational knots, 4-plats, 2-bridge links are defined similarly as above, but each component will have one min and max. 2-bridge knots were classified by Horst Schubert, using the fact that the 2-sheeted branched cover of the 3-sphere over the knot is a lens space, the names rational knot and rational link were coined by John Conway who defined them as arising from numerator closures of rational tangles. Horst Schubert, Über Knoten mit zwei Brücken, Mathematische Zeitschrift 65, louis H. Kauffman, Sofia Lambropoulou, On the classification of rational knots, L Enseignement Mathématique,49, 357–410. Adams, The Knot Book, An elementary introduction to the theory of knots. American Mathematical Society, Providence, RI,2004, xiv+307 pp. ISBN 0-8218-3678-1 Table and invariants of rational knots with up to 16 crossings