1.
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
2.
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
3.
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
4.
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
5.
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
6.
Linking number
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In mathematics, the linking number is a numerical invariant that describes the linking of two closed curves in three-dimensional space. Intuitively, the number represents the number of times that each curve winds around the other. The linking number is always an integer, but may be positive or negative depending on the orientation of the two curves, the linking number was introduced by Gauss in the form of the linking integral. Any two closed curves in space, if allowed to pass through themselves but not each other and this determines the linking number, Each curve may pass through itself during this motion, but the two curves must remain separated throughout. This is formalized as regular homotopy, which requires that each curve be an immersion. However, this condition does not change the definition of linking number. This fact is most easily proven by placing one circle in standard position, in detail, A single curve is regular homotopic to a standard circle. The complement of a circle is homeomorphic to a solid torus with a point removed. The fundamental group of 3-space minus a circle is the integers and this can be seen via the Seifert–Van Kampen theorem. Thus homotopy classes of a curve in 3-space minus a circle are determined by linking number and it is also true that regular homotopy classes are determined by linking number, which requires additional geometric argument. There is an algorithm to compute the number of two curves from a link diagram. Label each crossing as positive or negative, according to the following rule and that is, linking number = n 1 + n 2 − n 3 − n 42 where n1, n2, n3, n4 represent the number of crossings of each of the four types. The two sums n 1 + n 3 and n 2 + n 4 are always equal, note that n 1 − n 4 involves only the undercrossings of the blue curve by the red, while n 2 − n 3 involves only the overcrossings. Any two unlinked curves have linking number zero, however, two curves with linking number zero may still be linked. Reversing the orientation of either of the curves negates the linking number, the linking number is chiral, taking the mirror image of link negates the linking number. The convention for positive linking number is based on a right-hand rule, the winding number of an oriented curve in the x-y plane is equal to its linking number with the z-axis. More generally, if either of the curves is simple, then the first homology group of its complement is isomorphic to Z, in this case, the linking number is determined by the homology class of the other curve. In physics, the number is an example of a topological quantum number
7.
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
8.
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
9.
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
10.
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
11.
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
12.
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
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.
Knot
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A knot is a method of fastening or securing linear material such as rope by tying or interweaving. It may consist of a length of one or several segments of rope, string, webbing, twine, strap, Knots have been the subject of interest for their ancient origins, their common uses, and the area of mathematics known as knot theory. There is a variety of knots, each with properties that make it suitable for a range of tasks. Some knots are used to attach the rope to other such as another rope, cleat, ring. Some knots are used to bind or constrict objects, decorative knots usually bind to themselves to produce attractive patterns. While some people can look at diagrams or photos and tie the illustrated knots, Knot tying skills are often transmitted by sailors, scouts, climbers, canyoners, cavers, arborists, rescue professionals, stagehands, fishermen, linemen and surgeons. The International Guild of Knot Tyers is a dedicated to the Promotion of Knot tying. Truckers in need of securing a load may use a truckers hitch, Knots can save spelunkers from being buried under rock. Many knots can also be used as tools, for example, the bowline can be used as a rescue loop. The diamond hitch was used to tie packages on to donkeys. In hazardous environments such as mountains, knots are very important, note the systems mentioned typically require carabineers and the use of multiple appropriate knots. These knots include the bowline, double figure eight, munter hitch, munter mule, prusik, autoblock, thus any individual who goes into a mountainous environment should have basic knowledge of knots and knot systems to increase safety and the ability to undertake activities such as rappelling. Knots can be applied in combination to produce objects such as lanyards. In ropework, the end of a rope is held together by a type of knot called a whipping knot. Many types of textiles use knots to repair damage, macrame, one kind of textile, is generated exclusively through the use of knotting, instead of knits, crochets, weaves or felting. Macramé can produce self-supporting three-dimensional textile structures, as well as flat work, Knots weaken the rope in which they are made. When knotted rope is strained to its point, it almost always fails at the knot or close to it. The bending, crushing, and chafing forces that hold a knot in place also unevenly stress rope fibers, the exact mechanisms that cause the weakening and failure are complex and are the subject of continued study
18.
Circle
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A circle is a simple closed shape in Euclidean geometry. The distance between any of the points and the centre is called the radius, a circle is a simple closed curve which divides the plane into two regions, an interior and an exterior. Annulus, the object, the region bounded by two concentric circles. Arc, any connected part of the circle, centre, the point equidistant from the points on the circle. Chord, a segment whose endpoints lie on the circle. Circumference, the length of one circuit along the circle, or the distance around the circle and it is a special case of a chord, namely the longest chord, and it is twice the radius. Disc, the region of the bounded by a circle. Lens, the intersection of two discs, passant, a coplanar straight line that does not touch the circle. Radius, a line segment joining the centre of the circle to any point on the circle itself, or the length of such a segment, sector, a region bounded by two radii and an arc lying between the radii. Segment, a region, not containing the centre, bounded by a chord, secant, an extended chord, a coplanar straight line cutting the circle at two points. Semicircle, an arc that extends from one of a diameters endpoints to the other, in non-technical common usage it may mean the diameter, arc, and its interior, a two dimensional region, that is technically called a half-disc. A half-disc is a case of a segment, namely the largest one. Tangent, a straight line that touches the circle at a single point. The word circle derives from the Greek κίρκος/κύκλος, itself a metathesis of the Homeric Greek κρίκος, the origins of the words circus and circuit are closely related. The circle has been known since before the beginning of recorded history, natural circles would have been observed, such as the Moon, Sun, and a short plant stalk blowing in the wind on sand, which forms a circle shape in the sand. The circle is the basis for the wheel, which, with related inventions such as gears, in mathematics, the study of the circle has helped inspire the development of geometry, astronomy and calculus. Some highlights in the history of the circle are,1700 BCE – The Rhind papyrus gives a method to find the area of a circular field. The result corresponds to 256/81 as a value of π.300 BCE – Book 3 of Euclids Elements deals with the properties of circles
19.
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
20.
Unknotting problem
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In mathematics, the unknotting problem is the problem of algorithmically recognizing the unknot, given some representation of a knot, e. g. a knot diagram. There are several types of unknotting algorithms, a major unresolved challenge is to determine if the problem admits a polynomial time algorithm, that is, whether the problem lies in the complexity class P. First steps toward determining the computational complexity were undertaken in proving that the problem is in larger complexity classes, by using normal surfaces to describe the Seifert surfaces of a given knot, Hass, Lagarias & Pippenger showed that the unknotting problem is in the complexity class NP. Hara, Tani & Yamamoto claimed the result that unknotting is in AM ∩ co-AM. In 2011, Greg Kuperberg proved that the problem is in co-NP. The unknotting problem has the same complexity as testing whether an embedding of an undirected graph in Euclidean space is linkless. The Ochiai unknot featuring 139 vertices, for example, was originally unknotted by computer in 108 hours, several algorithms solving the unknotting problem are based on Hakens theory of normal surfaces, Hakens algorithm uses the theory of normal surfaces to find a disk whose boundary is the knot. Haken originally used this algorithm to show that unknotting is decidable, therefore, vertex enumeration methods can be used to list all of the extreme rays and test whether any of them corresponds to a bounding disk of the knot. The algorithm of Birman & Hirsch uses braid foliations, a different type of structure than a normal surface. However to analyze its behavior they return to normal surface theory, other approaches include, The number of Reidemeister moves needed to change an unknot diagram to the standard unknot diagram is at most polynomial in the number of crossings. Therefore, a brute force search for all sequences of Reidemeister moves can detect unknottedness in exponential time, similarly, any two triangulations of the same knot complement may be connected by a sequence of Pachner moves of length at most doubly exponential in the number of crossings. The time for this method would be triply exponential, however, experimental evidence suggests that this bound is very pessimistic, any arc-presentation of an unknot can be monotonically simplified to a minimal one using elementary moves. So a brute force search among all arc-presentations of not greater complexity gives an algorithm for the unknotting problem. Residual finiteness of the group gives an algorithm, check if the group has non-cyclic finite group quotient. This idea is used in Kuperbergs result that the problem is in co-NP. Knot Floer homology of the knot detects the genus of the knot, a combinatorial version of knot Floer homology allows it to be computed. Khovanov homology detects the unknot according to a result of Kronheimer, the complexity of Khovanov homology at least as high as the #P-hard problem of computing the Jones polynomial, but it may be calculated in practice using an algorithm and program of Bar-Natan. Understanding the complexity of algorithms is an active field of study
21.
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
22.
Bight (knot)
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In knot tying, a bight is a curved section or slack part between the two ends of a rope, string, or yarn. Any section of line that is bent into a U-shape is a bight, an open loop is a curve in a rope narrower than a bight but with separated ends. The term is used in a more specific way when describing Turks head knots. In order to make a knot, a bight must be passed. This slipped form of the knot is more easily untied, the traditional bow knot used for tying shoelaces is simply a reef knot with the final overhand knot made with two bights instead of the ends. Similarly, a hitch is a slipped clove hitch. The phrase in the means a bight of line is itself being used to make a knot. Specifically this means that the knot can be formed without access to the ends of the rope. This can be an important property for knots to be used in situations where the ends of the rope are inaccessible, many knots normally tied with an end also have a form which is tied in the bight. In other cases, a knot being tied in the bight is a matter of the method of tying rather than a difference in the form of the knot. For example, the hitch can be made in the bight if it is being slipped over the end of a post but not if being cast onto a closed ring. Other knots, such as the knot, cannot be tied in the bight without changing their final form
23.
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
24.
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
25.
Mutation (knot theory)
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In the mathematical field of knot theory, a mutation is an operation on a knot that can produce different knots. Suppose K is a given in the form of a knot diagram. Consider a disc D in the plane of the diagram whose boundary circle intersects K exactly four times. We may suppose that the disc is round and the four points of intersection on its boundary with K are equally spaced. The part of the knot inside the disc is a tangle, there are two reflections that switch pairs of endpoints of the tangle. There is also a rotation that results from composition of the reflections, a mutation replaces the original tangle by a tangle given by any of these operations. The result will always be a knot and is called a mutant of K. Mutants can be difficult to distinguish as they have a number of the same invariants and they have the same hyperbolic volume, and have the same HOMFLY polynomials. Conway and Kinoshita-Terasaka mutant pair, distinguished as knot genus 3 and 2, colin Adams, The Knot Book, American Mathematical Society, ISBN 0-8050-7380-9
26.
Cyclic group
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In algebra, a cyclic group or monogenous group is a group that is generated by a single element. Each element can be written as a power of g in multiplicative notation and this element g is called a generator of the group. Every infinite cyclic group is isomorphic to the group of Z. Every finite cyclic group of n is isomorphic to the additive group of Z/nZ. Every cyclic group is a group, and every finitely generated abelian group is a direct product of cyclic groups. A group G is called if there exists an element g in G such that G = ⟨g⟩ =. Since any group generated by an element in a group is a subgroup of that group, for example, if G = is a group of order 6, then g6 = g0, and G is cyclic. In fact, G is essentially the same as the set with addition modulo 6, for example,1 +2 ≡3 corresponds to g1 · g2 = g3, and 2 +5 ≡1 corresponds to g2 · g5 = g7 = g1, and so on. One can use the isomorphism χ defined by χ = i, the name cyclic may be misleading, it is possible to generate infinitely many elements and not form any literal cycles, that is, every gn is distinct. A group generated in this way is called a cyclic group. The French mathematicians known as Nicolas Bourbaki referred to a group as a monogenous group. The set of integers, with the operation of addition, forms a group and it is an infinite cyclic group, because all integers can be written as a finite sum or difference of copies of the number 1. In this group,1 and −1 are the only generators, every infinite cyclic group is isomorphic to this group. For every positive n, the set of integers modulo n, again with the operation of addition, forms a finite cyclic group. An element g is a generator of this group if g is relatively prime to n, thus, the number of different generators is φ, where φ is the Euler totient function, the function that counts the number of numbers modulo n that are relatively prime to n. Every finite cyclic group is isomorphic to a group Z/n, where n is the order of the group, the integer and modular addition operations, used to define the cyclic groups, are both the addition operations of commutative rings, also denoted Z and Z/n. If p is a prime, then Z/p is a finite field, every field with p elements is isomorphic to this one. For every positive n, the subset of the integers modulo n that are relatively prime to n, with the operation of multiplication
27.
Homeomorphism
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In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same. The word homeomorphism comes from the Greek words ὅμοιος = similar and μορφή = shape, roughly speaking, a topological space is a geometric object, and the homeomorphism is a continuous stretching and bending of the object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a torus are not. A function f, X → Y between two spaces and is called a homeomorphism if it has the following properties, f is a bijection, f is continuous. A function with three properties is sometimes called bicontinuous. If such a function exists, we say X and Y are homeomorphic, a self-homeomorphism is a homeomorphism of a topological space and itself. The homeomorphisms form a relation on the class of all topological spaces. The resulting equivalence classes are called homeomorphism classes, the open interval is homeomorphic to the real numbers R for any a < b. The unit 2-disc D2 and the square in R2 are homeomorphic. An example of a mapping from the square to the disc is, in polar coordinates. The graph of a function is homeomorphic to the domain of the function. A differentiable parametrization of a curve is an homeomorphism between the domain of the parametrization and the curve, a chart of a manifold is an homeomorphism between an open subset of the manifold and an open subset of a Euclidean space. The stereographic projection is a homeomorphism between the sphere in R3 with a single point removed and the set of all points in R2. If G is a group, its inversion map x ↦ x −1 is a homeomorphism. Also, for any x ∈ G, the left translation y ↦ x y, the right translation y ↦ y x, rm and Rn are not homeomorphic for m ≠ n. The Euclidean real line is not homeomorphic to the circle as a subspace of R2, since the unit circle is compact as a subspace of Euclidean R2. The third requirement, that f −1 be continuous, is essential, consider for instance the function f, [0, 2π) → S1 defined by f =
28.
Solid torus
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In mathematics, a solid torus is the topological space formed by sweeping a disk around a circle. It is homeomorphic to the Cartesian product S1 × D2 of the disk, a standard way to visualize a solid torus is as a toroid, embedded in 3-space. The solid torus is a connected, compact, orientable 3-dimensional manifold with boundary, the boundary is homeomorphic to S1 × S1, the ordinary torus. Since the disk D2 is contractible, the torus has the homotopy type of a circle. Therefore the fundamental group and homology groups are isomorphic to those of the circle, hyperbolic Dehn surgery Reeb foliation Whitehead manifold
29.
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
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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
31.
Godfried Toussaint
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Godfried T. Toussaint is a Professor of Computer Science and the Head of the Computer Science Program at New York University Abu Dhabi in Abu Dhabi, United Arab Emirates. Other interests include meander, compass and straightedge constructions, instance-based learning, music information retrieval and he is a co-founder of the Annual ACM Symposium on Computational Geometry, and the annual Canadian Conference on Computational Geometry. Along with Selim Akl, he is an author and namesake of the efficient Akl–Toussaint algorithm for the construction of the hull of a planar point set. This algorithm exhibits a computational complexity with expected value linear in the size of the input, three other well known proximity graphs are the nearest neighbor graph, the Urquhart graph, and the Gabriel graph. The first is contained in the spanning tree, and the Urquhart graph contains the RNG. Since all these graphs are nested together they are referred to as the Toussaint hierarchy and he recently spent a year in the Music Department at Harvard University doing research on musical similarity, a branch of music cognition. Since 2005 he has also been a researcher in the Centre for Interdisciplinary Research in Music Media and he applies computational geometric and discrete mathematics methods to the analysis of symbolically represented music in general, and rhythm in particular. In 2004 he discovered that the Euclidean algorithm for computing the greatest common divisor of two numbers implicitly generates almost all the most important traditional rhythms of the world and his application of mathematical methods for tracing the roots of Flamenco music were the focus of two Canadian television programs. In 1978 he was the recipient of the Pattern Recognition Societys Best Paper of the Year Award, in 1985 he was awarded a two-year Izaak Walton Killam Senior Research Fellowship by the Canada Council for the Arts. In 1988 he received an Advanced Systems Institute Fellowship from the British Columbia Advanced Systems Institute, in 1995 he was given the Vice-Chancellors Research Best-Practice Fellowship by the University of Newcastle in Australia. In 1996 he won the Canadian Image Processing and Pattern Recognition Societys Service Award for his contribution to research. In May 2001 he was honored with the David Thomson Award for excellence in graduate supervision, in 2009 he won a Radcliffe Fellowship from the Radcliffe Institute for Advanced Study at Harvard University to carry out a research project on the phylogenetics of the musical rhythms of the world. G. T. Toussaint, The Geometry of Musical Rhythm, Chapman and Hall/CRC, G. T. Toussaint, Computational Geometry, Editor, North-Holland Publishing Company, Amsterdam,1985. G. T. Toussaint, Computational Morphology, Editor, North-Holland Publishing Company, gassend, J. ORourke, and G. T. Toussaint, “All polygons flip finitely. Surveys on Discrete and Computational Geometry, Twenty Years Later, J. E. Goodman, J. Pach, J. ORourke and G. T. Toussaint, Pattern recognition, Chapter 51 in the Handbook of Discrete and Computational Geometry, Eds. J. E. Goodman and J. ORourke, Chapman & Hall/CRC, New York,2004, J. A. Calvo, K. Millett, and E. Rawdon, American Mathematical Society, Contemporary Mathematics Vol.304,2002, pp. 269–285. J. ORourke and G. T. Toussaint, Pattern recognition, Chapter 43 in the Handbook of Discrete and Computational Geometry, J. E. Goodman and J. ORourke, CRC Press, New York,1997, pp. 797–813. G. T. Toussaint, “A graph-theoretical primal sketch, ” in Computational Morphology, G. T. Toussaint, G. T. Toussaint, “Movable separability of sets, ” in Computational Geometry, G. T. Toussaint, Ed
32.
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
33.
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
34.
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
