In mathematics, knot theory is the area of topology that studies mathematical knots, which are defined as embeddings of a circle in 3-dimensional Euclidean space, R³. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of In Mathematics, a knot is an Embedding of a Circle in 3-dimensional Euclidean space, R 3 considered up to continuous deformations In Mathematics, an embedding (or imbedding) is one instance of some Mathematical structure contained within another instance such as a group Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the This is basically equivalent to a conventional knotted string with the ends joined together to prevent it from becoming undone. KNOT (1450 AM) is a commercial Classic Country music Radio station in Prescott Arizona, broadcasting to the Flagstaff - Prescott Two mathematical knots are equivalent if one can be transformed into the other via a deformation of R³ upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself. In the mathematical subject of Topology, an ambient isotopy, also called an h-isotopy is a kind of continuous distortion of an "ambient space" a Manifold

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 common method of describing a knot is a planar diagram called a knot diagram. But any given knot can be drawn in many different ways using a knot diagram. Therefore, a fundamental problem in knot theory is determining when two descriptions represent the same knot. One way of distinguishing knots is by using a knot invariant, a "quantity" which remains the same even with different descriptions of a knot. In the mathematical field of Knot theory, a knot invariant is a quantity (in a broad sense defined for each knot which is the same for equivalent knots

The concept of a knot has also been extended to higher dimensions by considering n-dimensional spheres in m-dimensional Euclidean space. In Mathematics, an n -sphere is a generalization of an ordinary Sphere to arbitrary Dimension.

Contents 1 History 2 Knot equivalence 3 Knot diagrams 3.1 Reidemeister moves 4 Knot invariants 4.1 Knot polynomials 4.2 Hyperbolic invariants 5 Higher dimensions 6 Adding knots 7 Tabulating knots 7.1 Alexander-Briggs notation 7.2 The Dowker notation 7.3 Conway notation 8 See also 9 References 10 Further reading 11 External links 11.1 History 11.2 Knot tables and software

History

Main article: History of knot theory

Research in knot theory began with the creation of knot tables and the systematic tabulation of knots. For thousands of years knots have been used for basic purposes such as recording information, fastening and tying objects together Ever since Sir William Thomson 's Vortex theory, mathematicians have tired to classify and tabulate all possible knots. While tabulation remains an important task, today's researchers have a wide variety of backgrounds and goals. Classical knot theory, as initiated by Max Dehn, J. W. Alexander, and others, is primarily concerned with the knot group and invariants from homology theory such as the Alexander polynomial. Max Dehn ( November 13, 1878, Hamburg, Germany – June 27, 1952, Black Mountain, North Carolina, James Waddell Alexander II ( September 19, 1888 – September 23, 1971) was an important Topologist of the pre-WWII era and part of In Mathematics, a knot is an Embedding of a Circle into 3-dimensional Euclidean space. In Mathematics, homology theory is the Axiomatic study of the intuitive geometric idea of homology of cycles on Topological spaces It can be broadly In Mathematics, the Alexander polynomial is a Knot invariant which assigns a Polynomial with integer coefficients to each knot type

In the late 1970s, William Thurston introduced hyperbolic geometry into the study of knots with the hyperbolization theorem. William Paul Thurston (born October 30, 1946) is an American Mathematician. In Thurston's geometrization conjecture states that compact 3-manifolds can be decomposed into Submanifolds that have geometric structures Many knots were shown to be hyperbolic knots, enabling the use of geometry in defining new, powerful knot invariants. In Mathematics, a hyperbolic link is a link in the 3-sphere with complement that has a complete Riemannian metric of constant negative In the mathematical field of Knot theory, a knot invariant is a quantity (in a broad sense defined for each knot which is the same for equivalent knots

The discovery of the Jones polynomial by Vaughan Jones in 1984 (Sossinsky 2002, p. In the mathematical field of Knot theory, the Jones polynomial is a Knot polynomial discovered by Vaughan Jones in 1983 Vaughan Frederick Randal Jones DCNZM (born 31 December 1952) is a New Zealand Mathematician, known for his work on  71-89), and subsequent contributions from Edward Witten, Maxim Kontsevich, and others, revealed deep connections between knot theory and mathematical methods in statistical mechanics and quantum field theory . Edward Witten (born August 26, 1951) is an American Theoretical physicist and Professor at the Institute for Advanced Study Maxim Lvovich Kontsevich (Максим Львович Концевич (born August 25, 1964) is a Russian Mathematician. Statistical mechanics is the application of Probability theory, which includes mathematical tools for dealing with large populations to the field of Mechanics In quantum field theory (QFT the forces between particles are mediated by other particles A plethora of knot invariants have been invented since then, utilizing sophisticated tools such as quantum groups and Floer homology. In Mathematics and Theoretical physics, quantum groups are certain Noncommutative algebras that first appeared in the theory of Quantum integrable systems Floer homology is a mathematical tool used in the study of Symplectic geometry and low-dimensional Topology.

In the last several decades of the 20th century, scientists and mathematicians began finding applications of knot theory to problems in biology and chemistry. Foundations of modern biology There are five unifying principles Chemistry (from Egyptian kēme (chem meaning "earth") is the Science concerned with the composition structure and properties Knot theory can be used to determine if a molecule is chiral (has a "handedness") or not. The term chiral (pronounced /ˈkaɪɹ(əl̩/ is used to describe an object that is non- superimposable on its mirror image The closely related theory of tangles have been effectively used in studying the action of certain enzymes on DNA. In Mathematics, an n - tangle is a proper embedding of the disjoint union of n arcs into a 3-ball. (Flapan 2000) There is growing interest in the study of physical knots in order to understand knotting phenomena in DNA and polymers. Physical knot theory is the study of Mathematical models of Knotting phenomena often motivated by physical considerations from Biology, Chemistry Knot theoretic topology may be crucial in the construction of quantum computers, through the model of topological quantum computation. A topological quantum computer is a theoretical Quantum computer that employs two-dimensional Quasiparticles called Anyons whose World lines cross

Knot equivalence

A knot is created by beginning with a one-dimensional line segment, wrapping it around itself arbitrarily, and then fusing its two free ends together to form a closed loop. In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it When mathematical topologists consider knots and other entanglements such as links and braids, they describe how the knot is positioned in the space around it, called the ambient space. In Mathematics, a link is a collection of knots which do not intersect but which may be linked (or knotted together In Topology, braid theory is an abstract geometric Theory studying the everyday Braid concept and some generalisations An ambient space, ambient configuration space, or electroambient space, is the space surrounding an object. If the knot can be moved smoothly, without cutting or passing a segment through another, until it coincides with another knot, the two knots are considered equivalent. The idea of knot equivalence is to give a precise definition of when two embeddings should be considered the same.

The basic problem of knot theory, the recognition problem, can thus be stated: given two knots, determine whether they are equivalent or not. Algorithms exist to solve this problem, with the first given by Wolfgang Haken. In Mathematics, Computing, Linguistics and related subjects an algorithm is a sequence of finite instructions often used for Calculation Nonetheless, these algorithms use very many steps, and a major issue in the theory is to understand how hard this problem really is (Hass 1997). The special case of recognizing the unknot, called the unknotting problem, is of particular interest. The unknot arises in the mathematical theory of knots. Intuitively the unknot is a closed loop of rope without a Knot in it In Mathematics, the unknotting problem is the problem of algorithmically recognizing the Unknot, given some input e

Knot diagrams

A useful way to visualise and manipulate knots is to project the knot onto a plane—think of the knot casting a shadow on the wall. A small perturbation in the choice of projection will ensure that it is one-to-one except at the double points, called crossings, where the "shadow" of the knot crosses itself once transversely (Rolfsen 1976). At each crossing we must indicate which section is "over" and which is "under", so as to be able to recreate the original knot. This is often done by creating a break in the strand going underneath. If by following the diagram the knot alternately crosses itself "over" and "under", then the diagram represents an alternating knot. In Knot theory, a link diagram is alternating if the crossings alternate under over under over as you travel along each component of the link

Reidemeister moves

Main article: Reidemeister move

In 1927, working with this diagrammatic form of knots, J.W. Alexander and G. In the mathematical area of Knot theory, a Reidemeister move refers to one of three local moves on a Link diagram. James Waddell Alexander II ( September 19, 1888 – September 23, 1971) was an important Topologist of the pre-WWII era and part of B. Briggs, and independently Kurt Reidemeister, demonstrated that two knot diagrams belonging to the same knot can be related by a sequence of three kinds of moves on the diagram, shown below. Kurt Werner Friedrich Reidemeister ( October 13, 1893 - July 8, 1971) was a Mathematician born in Braunschweig (Brunswick These operations, now called the Reidemeister moves, are:

1. Twist and untwist in either direction.
2. Move one strand completely over another.
3. Move a strand completely over or under a crossing.

Reidemeister moves

Type I	Type II
Type III

Knot invariants

Main article: knot invariant

A knot invariant is a "quantity" that is the same for equivalent knots (Adams 2001, Lickorish 1997, Rolfsen 1976). In the mathematical field of Knot theory, a knot invariant is a quantity (in a broad sense defined for each knot which is the same for equivalent knots An invariant may take the same value on two different knots, so by itself may be incapable of distinguishing all knots. An elementary invariant is tricolorability. The tricolorability of a Knot refers to the ability of a knot to be colored with three colors according to two rules

"Classical" knot invariants include the knot group, which is the fundamental group of the knot complement, and the Alexander polynomial, which can be computed from the Alexander invariant, a module constructed from the infinite cyclic cover of the knot complement (Lickorish 1997, Rolfsen 1976). In Mathematics, a knot is an Embedding of a Circle into 3-dimensional Euclidean space. In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology. In Mathematics, the knot complement of a tame knot K is the set-theoretic complement of the interior of the embedding of a Solid torus In Mathematics, the Alexander polynomial is a Knot invariant which assigns a Polynomial with integer coefficients to each knot type In the late 20th century, invariants such as "quantum" knot polynomials and hyperbolic invariants were discovered. These aforementioned invariants are only the tip of the iceberg of modern knot theory.

Knot polynomials

Main article: knot polynomial

A knot polynomial is a knot invariant that is a polynomial. In the mathematical field of Knot theory, a knot polynomial is a Knot invariant in the form of a Polynomial whose coefficients encode some of In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations Well-known examples include the Jones and Alexander polynomials. In the mathematical field of Knot theory, the Jones polynomial is a Knot polynomial discovered by Vaughan Jones in 1983 In Mathematics, the Alexander polynomial is a Knot invariant which assigns a Polynomial with integer coefficients to each knot type A variant of the Alexander polynomial, the Alexander-Conway polynomial, is a polynomial in the variable z with integer coefficients (Lickorish 1997). In Mathematics, the Alexander polynomial is a Knot invariant which assigns a Polynomial with integer coefficients to each knot type The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French

Suppose we are given a link diagram which is oriented, i. e. every component of the link has a preferred direction indicated by an arrow. Also suppose L ₊ ,L ₋ ,L₀ are oriented link diagrams resulting from changing the diagram at a specified crossing of the diagram, as indicated in the figure:

Then the Alexander-Conway polynomial, C(z), is recursively defined according to the rules:

• C(O) = 1 (where O is any diagram of the unknot)
• C(L ₊ ) = C(L ₋ ) + zC(L₀)

The second rule is what is often referred to as a skein relation. The unknot arises in the mathematical theory of knots. Intuitively the unknot is a closed loop of rope without a Knot in it A central question in the mathematical theory of knots is whether two Knot diagrams represent the same knot To check that these rules give an invariant, one should determine that the polynomial does not change under the three Reidemeister moves. Many important knot polynomials can be defined in this way.

The following is an example of a typical computation using a skein relation. It computes the Alexander-Conway polynomial of the trefoil knot. In Knot theory, the trefoil knot is the simplest nontrivial knot. The yellow patches indicate where we applied the relation.

C()=C() + z C()

gives the unknot and the Hopf link. In mathematical Knot theory, the Hopf link, named after Heinz Hopf, is the simplest nontrivial link with more than one component Applying the relation to the Hopf link where indicated,

C() = C() + z C()

gives a link deformable to one with 0 crossings (it is actually the unlink of two components) and an unknot. In the mathematical field of Knot theory, the unlink is a link that is equivalent (under Ambient isotopy) to finitely many disjoint circles in The unlink takes a bit of sneakiness:

C() = C()+ z C()

which implies that C(unlink of two components) = 0, since the first two polynomials are of the unknot and thus equal.

Putting all this together will show:

C(trefoil) = 1 + z (0 + z) = 1 + z²

Since the Alexander-Conway polynomial is a knot invariant, this shows that the trefoil is not equivalent to the unknot. So the trefoil really is "knotted".

The left handed trefoil knot.

The right handed trefoil knot.

Actually, there are two trefoil knots, called the right and left-handed trefoils, which are mirror images of each other (take a diagram of the trefoil given above and change each crossing to the other way to get the mirror image). In the mathematical field of Knot theory, a chiral knot is a knot that is not equivalent to its mirror image These are not equivalent to each other! This was shown by Max Dehn, before the invention of knot polynomials, using group theoretical methods (Dehn 1914). Max Dehn ( November 13, 1878, Hamburg, Germany – June 27, 1952, Black Mountain, North Carolina, But the Alexander-Conway polynomial of each kind of trefoil will be the same, as can be seen by going through the computation above with the mirror image. The Jones polynomial can in fact distinguish between the left and right-handed trefoil knots (Lickorish 1997).

Hyperbolic invariants

William Thurston proved many knots are hyperbolic knots, meaning that the knot complement, i. William Paul Thurston (born October 30, 1946) is an American Mathematician. In Mathematics, a hyperbolic link is a link in the 3-sphere with complement that has a complete Riemannian metric of constant negative In Mathematics, the knot complement of a tame knot K is the set-theoretic complement of the interior of the embedding of a Solid torus e. the points of 3-space not on the knot, admit a geometric structure, in particular that of hyperbolic geometry. In The hyperbolic structure depends only on the knot so any quantity computed from the hyperbolic structure is then a knot invariant. (Adams 2001)

Geometry lets us visualize what the inside of a knot or link complement looks like by imagining light rays as traveling along the geodesics of the geometry. An example is provided by the picture of the complement of the Borromean rings. In Mathematics, the Borromean rings consist of three topological Circles which are linked and form a Brunnian link, i The inhabitant of this link complement is viewing the space from near the red component. The balls in the picture are views of horoball neighborhoods of the link. In Mathematics, particularly Hyperbolic geometry, a horoball is a specific kind of n -dimensional object in hyperbolic ''n''-space. By thickening the link in a standard way, we obtain what are called horoball neighborhoods of the link components. Even though the boundary of a neighborhood is a torus, when viewed from inside the link complement, it looks like a sphere. Each link component shows up as infinitely many spheres (of one color) as there are infinitely many light rays from the observer to the link component. The fundamental parallelogram (which is indicated in the picture), tiles both vertically and horizontally and shows how to extend the pattern of spheres infinitely.

This pattern, the horoball pattern, is itself a useful invariant. Other hyperbolic invariants include the shape of the fundamental paralleogram, length of shortest geodesic, and volume. Modern knot and link tabulation efforts have utilized these invariants effectively. Fast computers and clever methods of obtaining these invariants make calculating these invariants, in practice, a simple task. (Adams, Hildebrand, & Weeks, 1991)

Higher dimensions

In four dimensions, any closed loop of one-dimensional string is equivalent to an unknot. We can achieve the necessary deformation in two steps. The first step is to "push" the loop into a three-dimensional subspace, which is always possible, though technical to explain. The second step is changing crossings. Suppose one strand is behind another as seen from a chosen point. Lift it into the fourth dimension, so there is no obstacle (the front strand having no component there); then slide it forward, and drop it back, now in front. An analogy for the plane would be lifting a string up off the surface.

Since a knot can be considered topologically a 1-dimensional sphere, the next generalization is to consider a two dimensional sphere embedded in a four dimensional sphere. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe Such an embedding is unknotted if there is a homeomorphism of the 4-sphere onto itself taking the 2-sphere to a standard "round" 2-sphere. Suspended knots and spun knots are two typical families of such 2-sphere knots.

The mathematical technique called "general position" implies that for a given n-sphere in the m-sphere, if m is large enough (depending on n), the sphere should be unknotted. In general, piecewise-linear n-spheres form knots only in (n+2)-space (Christopher Zeeman 1963), although this is no longer a requirement for smoothly knotted spheres. Piecewise linear may refer to Piecewise linear function Piecewise linear manifold In Mathematics, an n -sphere is a generalization of an ordinary Sphere to arbitrary Dimension. Sir Erik Christopher Zeeman FRS (born 4 February 1925) is a Japanese born British Mathematician known for his work in In fact, there are smoothly knotted 4k-1-spheres in 6k-space, e. g. there is a smoothly knotted 3-sphere in the 6-sphere (Haefliger 1962, Levine 1965). Thus the codimension of a smooth knot can be arbitrarily large when not fixing the dimension of the knotted sphere; however, any smooth k-sphere in an n-sphere with 2n-3k-3 > 0 is unknotted. The notion of a knot has further generalisations in mathematics, see: knot (mathematics). In Mathematics, a knot is an Embedding of a Circle in 3-dimensional Euclidean space, R 3 considered up to continuous deformations

Adding knots

Main article: knot sum

Two knots can be added by cutting both knots and joining the pairs of ends. The operation is called the knot sum, or sometimes the connected sum or composition of two knots. This can be formally defined as follows (Adams 2001): consider a planar projection of each knot and suppose these projections are disjoint. Find a rectangle in the plane where one pair of opposite sides are arcs along each knot while the rest of the rectangle is disjoint from the knots. Form a new knot by deleting the first pair of opposite sides and adjoining the other pair of opposite sides. The resulting knot is a sum of the original knots. It is possible to obtain at most two different knots in this manner, although this ambiguity can be eliminated regarding the knots as oriented and utilizing an oriented sum.

The knot sum of oriented knots is commutative and associative. A surface S in the Euclidean space R 3 is orientable if a two-dimensional figure (for example) cannot be moved around the surface and back In Mathematics, commutativity is the ability to change the order of something without changing the end result In Mathematics, associativity is a property that a Binary operation can have There is also a prime decomposition for a knot which allows us to define a prime or composite knot, analogous to prime and composite numbers (Schubert, 1949). In Knot theory, a prime knot is a knot which is in a certain sense indecomposable In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 For oriented knots, this decomposition is also unique. Higher dimensional knots can also be added but there are some differences. While you cannot form the unknot in three dimensions by adding two non-trivial knots, you can in higher dimensions, at least when one considers smooth knots in codimension at least 3.

Tabulating knots

Traditionally, knots have been catalogued in terms of crossing number. In Mathematics, crossing numbers arise in two related contexts in Knot theory and in Graph theory. The number of nontrivial knots of a given crossing number increases rapidly, making tabulation computationally difficult. Knot tables generally include only prime knots and only one entry for a knot and its mirror image (even if they are different). The sequence of the number of prime knots of a given crossing number, up to crossing number 16, is 0, 0, 0, 1, 1, 2, 3, 7, 21, 49, 165, 552, 2176, 9988, 46972, 253293, 1388705. . . (sequence A002863 in OEIS). The On-Line Encyclopedia of Integer Sequences ( OEIS) also cited simply as Sloane's, is an extensive searchable Database of Integer sequences While exponential upper and lower bounds for this sequence are known, it has not been proven that this sequence is strictly increasing (Adams 2001).

The first knot tables by Tait, Little, and Kirkman used knot diagrams, although Tait also used a precursor to the Dowker notation. In the mathematical field of Knot theory, the Dowker notation, also called the Dowker-Thistlethwaite notation or code for a knot is a sequence Different notations have been invented for knots which allow more efficient tabulation.

The early tables attempted to list all knots of at most 10 crossings, and all alternating knots of 11 crossings. The development of knot theory due to Alexander, Reidemeister, Seifert, and others eased the task of verification and tables of knots up to and including 9 crossings were published by Alexander-Briggs and Reidemeister in the late 1920s.

The first major verification of this work was done in the 1960s by John Horton Conway, who not only developed a new notation but also the Alexander-Conway polynomial (Conway 1970, Doll-Hoste 1991). John Horton Conway (born December 26, 1937, Liverpool, England) is a prolific mathematician active in the theory of finite groups In Mathematics, the Alexander polynomial is a Knot invariant which assigns a Polynomial with integer coefficients to each knot type This verified the list of knots of at most 11 crossings and a new list of links up to 10 crossings. Conway found a number of omissions but only one duplication in the Tait-Little tables; however he missed the duplicates called the Perko pair, which would only be noticed in 1974 by Kenneth Perko (Perko 1974). In the mathematical theory of knots, the Perko pair, named after Kenneth Perko is a pair of entries in classical knot tables that actually represent the same knot This famous error would propagate when Dale Rolfsen added a knot table in his influential text, based on Conway's work.

Alexander-Briggs notation

This is the most traditional notation, due to the 1927 paper of J. W. Alexander and G. Briggs and later extended by Dale Rolfsen in his knot table. The notation simply organizes knots by their crossing number. One writes the crossing number with a subscript to denote its order amongst all knots with that crossing number. This order is arbitrary and so has no special significance.

The Dowker notation

Main article: Dowker notation

The Dowker notation, also called the Dowker-Thistlethwaite notation or code, for a knot is a finite sequence of even integers. In the mathematical field of Knot theory, the Dowker notation, also called the Dowker-Thistlethwaite notation or code for a knot is a sequence The numbers are generated by following the knot and marking the crossings with consecutive integers. Since each crossing is visited twice, this creates a pairing of even integers with odd integers. An appropriate sign is given to indicate over and undercrossing. For example, in the figure the knot diagram has crossings labelled with the pairs (1,6) (3,−12) (5,2) (7,8) (9,−4) and (11,−10). The Dowker notation for this labelling is the sequence: 6 −12 2 8 −4 −10. A knot diagram has more than one possible Dowker notation, and there is a well-understood ambiguity when reconstructing a knot from a Dowker notation.

Conway notation

Main article: Conway notation (knot theory)

The Conway notation for knots and links, named after John Horton Conway, is based on the theory of tangles (Conway 1970). In Knot theory, Conway notation, invented by John Horton Conway, is a way of describing knots that allows one to see many of a knots properties from it John Horton Conway (born December 26, 1937, Liverpool, England) is a prolific mathematician active in the theory of finite groups In Mathematics, an n - tangle is a proper embedding of the disjoint union of n arcs into a 3-ball. The advantage of this notation is that it reflects some properties of the knot or link.

The notation describes how to construct a particular link diagram of the link. Start with a basic polyhedron, a 4-valent connected planar graph with no digon regions. In Geometry a digon is a degenerate Polygon with two sides (edges and two vertices. Such a polyhedron is denoted first by the number of vertices then a number of asterisks which determine the polyhedron's position on a list of basic polyhedron. For example, 10** denotes the second 10-vertex polyhedron on Conway's list.

Each vertex then has an algebraic tangle substituted into it (each vertex is oriented so there is no arbitrary choice in substitution). In Mathematics, an n - tangle is a proper embedding of the disjoint union of n arcs into a 3-ball. Each such tangle has a notation consisting of numbers and + or − signs.

An example is 1*2 −3 2. The 1* denotes the only 1-vertex basic polyhedron. The 2 −3 2 is a sequence describing the continued fraction associated to a rational tangle. In Mathematics, an n - tangle is a proper embedding of the disjoint union of n arcs into a 3-ball. One inserts this tangle at the vertex of the basic polyhedron 1*.

A more complicated example is 8*3. 1. 2 0. 1. 1. 1. 1. 1 Here again 8* refers to a basic polyhedron with 8 vertices. The periods separate the notation for each tangle.

Any link admits such a description, and it is clear this is a very compact notation even for very large crossing number. There are some further shorthands usually used. The last example is usually written 8*3:2 0, where we omitted the ones and kept the number of dots excepting the dots at the end. For an algebraic knot such as in the first example, 1* is often omitted.

Conway's pioneering paper on the subject lists up to 10-vertex basic polyhedra of which he uses to tabulate links, which have become standard for those links. For a further listing of higher vertex polyhedra, there are nonstandard choices available.

See also

• List of knot theory topics
• Braid theory
• Braid group
• Braided monoidal category
• Legendrian knots
• Knotane
• Topoisomerase

References

• Colin Adams, The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, 2001, ISBN 0-7167-4219-5
• Adams, Colin; Hildebrand, Martin; Weeks, Jeffrey; Hyperbolic invariants of knots and links. This list contains articles related to the mathematical theory of knots links and braids. In Topology, braid theory is an abstract geometric Theory studying the everyday Braid concept and some generalisations In Mathematics, the braid group on n strands, denoted by B n, is a certain group which has an intuitive geometrical representation In Mathematics, a braided monoidal category is a Monoidal category C equipped with a braiding; that is there is a Natural isomorphism In Mathematics, contact geometry is the study of a geometric structure on Smooth manifolds given by a hyperplane distribution in the Tangent bundle In Chemistry, a molecular knot (knotane is a mechanically-interlocked molecular architecture that is analogous to a macroscopic Knot. Topoisomerases (type I, type II) are Isomerase Enzymes that act on the topology of DNA. Colin Conrad Adams (born October 13, 1956) is a Mathematician primarily working in the areas of Hyperbolic 3-manifolds and Knot theory Year 2001 ( MMI) was a Common year starting on Monday according to the Gregorian calendar. Trans. Amer. Math. Soc. 326 (1991), no. 1, 1--56.
• Bar-Natan, Dror (1995), “On the Vassiliev knot invariants”, Topology 34 (2): 423--472 
• John Horton Conway, An enumeration of knots and links, and some of their algebraic properties. John Horton Conway (born December 26, 1937, Liverpool, England) is a prolific mathematician active in the theory of finite groups 1970 Computational Problems in Abstract Algebra (Proc. Conf. , Oxford, 1967) pp. 329--358 Pergamon, Oxford
• Max Dehn, Die beiden Kleeblattschlingen, Math. Ann. 75 (1914), 402-413.
• Helmut Doll and Jim Hoste, A tabulation of oriented links. With microfiche supplement. Math. Comp. 57 (1991), no. 196, 747--761.
• Flapan, Erica (2000), “When topology meets chemistry: A topological look at molecular chirality”, Outlooks (Cambridge University Press,Cambridge; Mathematical Association of America, Washington, DC), ISBN 0-521-66254-0  ISBN 0-521-66482-9 xiv+241 pp
• André Haefliger, Knotted (4k-1)-spheres in 6k-space. Ann. of Math. (2) 75 1962 452--466.
• Joel Hass, "Algorithms for recognizing knots and 3-mainifolds". arXiv:math.GT/9712269
• Jerome Levine, A classification of differentiable knots. The arXiv ( pronounced " Archive " as if the "X" were the Greek letter Chi, χ is an Archive for electronic Jerome Paul Levine ( May 4, 1937 – April 8, 2006) was a Mathematician who contributed to the understanding of Knot theory Ann. of Math. (2) 82 1965 15--50. The Annals of Mathematics (ISSN 0003-486X abbreviated as Ann of Math
• Kontsevich, Maxim (1993), “Vassiliev's knot invariants”, I. Maxim Lvovich Kontsevich (Максим Львович Концевич (born August 25, 1964) is a Russian Mathematician. M. Gelfand Seminar, Adv. Soviet Math. (Providence, RI: Amer. Math. Soc. ) 16: 137--150 
• W. B. Raymond Lickorish, An Introduction to Knot Theory, Graduate Texts in Mathematics, Springer, 1997, ISBN 0-387-98254-X
• Kenneth A. William Bernard Raymond Lickorish is a Mathematician. He is Emeritus professor of Geometric topology in DPMMS, University of Cambridge Perko Jr. , On the classification of knots. Proc. Amer. Math. Soc. 45 (1974), 262--266.
• Dale Rolfsen, Knots and Links, 1976, ISBN 0-914098-16-0
• Schubert, Horst (1949), “Die eindeutige Zerlegbarkeit eines Knotens in Primknoten”, Heidelberger Akad. Year 1976 ( MCMLXXVI) was a Leap year starting on Thursday (link will display full calendar of the Gregorian calendar. Wiss. Math. -Nat. Kl. (no. 3): 57--104 
• Silver, Dan, Scottish physics and knot theory's odd origins (expanded version of Silver, "Knot theory's odd origins," American Scientist, 94, No. 2, 158-165)
• Sossinsky, Alexei (2002), Knots, mathematics with a twist, Harvard University Press, ISBN 0-674-00944-4 
• Turaev, V. G. (1994), “Quantum invariants of knots and 3-manifolds”, de Gruyter Studies in Mathematics (Berlin: Walter de Gruyter & Co. ) 18, ISBN 3-11-013704-6 
• Witten, Edward (1989), “Quantum field theory and the Jones polynomial”, Comm. Edward Witten (born August 26, 1951) is an American Theoretical physicist and Professor at the Institute for Advanced Study Math. Phys. 121 (3): 351--399 
• E. C. Zeeman, Unknotting combinatorial balls. Sir Erik Christopher Zeeman FRS (born 4 February 1925) is a Japanese born British Mathematician known for his work in Ann. of Math. (2) 78 1963 501--526. The Annals of Mathematics (ISSN 0003-486X abbreviated as Ann of Math

Further reading

There are a number of introductions to knot theory. A classical introduction for graduate students or advanced undergraduates is Rolfsen (1976), given in the references. Other good texts from the references are Adams (2001) and Lickorish (1997). Adams is informal and accessible for the most part to high schoolers. Lickorish is a rigorous introduction for graduate students, covering a nice mix of classical and modern topics.

• Richard H. Crowell and Ralph Fox,Introduction to Knot Theory, 1977, ISBN 0-387-90272-4
• Gerhard Burde and Heiner Zieschang, Knots, De Gruyter Studies in Mathematics, 1985, Walter de Gruyter, ISBN 3-11-008675-1
• Louis H. Kauffman, On Knots, 1987, ISBN 0-691-08435-1

External links

History

• Thomson, Sir William (Lord Kelvin), On Vertex Atoms, Proceedings of the Royal Society of Edinburgh, Vol. Ralph Hartzler Fox ( March 24, 1913, Morrisville Pennsylvania &ndash December 23, 1973, Philadelphia) was an Also 1977 (album by Ash. Year 1977 ( MCMLXXVII) was a Common year starting on Saturday (link displays Heiner Zieschang ( 12 November 1936 in Kiel – 5 April 2004) was a German Mathematician. Louis H Kauffman ( 3 February, 1945) is an American Mathematician, topologist, and professor of Mathematics in the Department of Mathematics Year 1987 ( MCMLXXXVII) was a Common year starting on Thursday (link displays 1987 Gregorian calendar) VI, 1867, pp. 94-105.
• Silliman, Robert H. , William Thomson: Smoke Rings and Nineteenth-Century Atomism, Isis, Vol. 54, No. 4. (Dec. , 1963), pp. 461-474. JSTOR link
• Movie of a modern recreation of Tait's smoke ring experiment

Knot tables and software

• someworks related with Knots especially other string-cycle and knot-like patterns in Falk Arts
• KnotInfo: Table of Knot Invariants and Knot Theory Resources
• The wiki Knot Atlas - detailed info on individual knots in knot tables
• KnotPlot - software to investigate geometric properties of knots

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