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Fáry–Milnor theorem

In the mathematical theory of knots, the Fáry–Milnor theorem, named after István Fáry and John Milnor, states that three-dimensional smooth curves with small total curvature must be unknotted. The the

Hyperbolic volume

In the mathematical field of knot theory, the hyperbolic volume of a hyperbolic link is the volume of the link's complement with respect to its complete hyperbolic metric. The volume is necessarily a

Möbius energy

In mathematics, the Möbius energy of a knot is a particular knot energy, i.e., a functional on the space of knots. It was discovered by Jun O'Hara, who demonstrated that the energy blows up as the kno

Knots Unravelled

Knots Unravelled: From String to Mathematics is a book on the mathematics of knots, intended for schoolchildren and other non-mathematicians. It was written by mathematician Meike Akveld and mathemati

Conway notation (knot theory)

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 const

Jones polynomial

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 an oriented knot or link which assigns to e

Conway sphere

In mathematical knot theory, a Conway sphere, named after John Horton Conway, is a 2-sphere intersecting a given knot or link in a 3-manifold transversely in four points. In a knot diagram, a Conway s

Dehornoy order

In the mathematical area of braid theory, the Dehornoy order is a left-invariant total order on the braid group, found by Patrick Dehornoy. Dehornoy's original discovery of the order on the braid grou

Slice genus

In mathematics, the slice genus of a smooth knot K in S3 (sometimes called its Murasugi genus or 4-ball genus) is the least integer g such that K is the boundary of a connected, orientable 2-manifold

Birman–Wenzl algebra

In mathematics, the Birman–Murakami–Wenzl (BMW) algebra, introduced by Joan Birman and Hans Wenzl and Jun Murakami, is a two-parameter family of algebras of dimension having the Hecke algebra of the s

Biracks and biquandles

In mathematics, biquandles and biracks are sets with binary operations that generalize quandles and racks. Biquandles take, in the theory of virtual knots, the place that quandles occupy in the theory

Chord diagram (mathematics)

In mathematics, a chord diagram consists of a cyclic order on a set of objects, together with a one-to-one pairing (perfect matching) of those objects. Chord diagrams are conventionally visualized by

Writhe

In knot theory, there are several competing notions of the quantity writhe, or . In one sense, it is purely a property of an oriented link diagram and assumes integer values. In another sense, it is a

Kauffman polynomial

In knot theory, the Kauffman polynomial is a 2-variable knot polynomial due to Louis Kauffman. It is initially defined on a link diagram as , where is the writhe of the link diagram and is a polynomia

Physical knot theory

Physical knot theory is the study of mathematical models of knotting phenomena, often motivated by considerations from biology, chemistry, and physics (Kauffman 1991). Physical knot theory is used to

Chirality (mathematics)

In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations al

Eilenberg–Mazur swindle

In mathematics, the Eilenberg–Mazur swindle, named after Samuel Eilenberg and Barry Mazur, is a method of proof that involves paradoxical properties of infinite sums. In geometric topology it was intr

Knot tabulation

Ever since Sir William Thomson's vortex theory, mathematicians have tried to classify and tabulate all possible knots. As of May 2008, all prime knots up to 16 crossings have been tabulated. The major

HOMFLY polynomial

In the mathematical field of knot theory, the HOMFLY polynomial or HOMFLYPT polynomial, sometimes called the generalized Jones polynomial, is a 2-variable knot polynomial, i.e. a knot invariant in the

Quadrisecant

In geometry, a quadrisecant or quadrisecant line of a space curve is a line that passes through four points of the curve. This is the largest possible number of intersections that a generic space curv

List of mathematical knots and links

This article contains a list of mathematical knots and links. See also list of knots, list of geometric topology topics.

2-bridge knot

In the mathematical field of knot theory, a 2-bridge knot is a knot which can be regular isotoped so that the natural height function given by the z-coordinate has only two maxima and two minima as cr

Alternating planar algebra

The concept of alternating planar algebras first appeared in the work of Hernando Burgos-Soto on the Jones polynomial of . Alternating planar algebras provide an appropriate algebraic framework for ot

Alexander polynomial

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 polynomi

Alexander's theorem

In mathematics Alexander's theorem states that every knot or link can be represented as a closed braid; that is, a braid in which the corresponding ends of the strings are connected in pairs. The theo

Linkless embedding

In topological graph theory, a mathematical discipline, a linkless embedding of an undirected graph is an embedding of the graph into three-dimensional Euclidean space in such a way that no two cycles

Cyclic surgery theorem

In three-dimensional topology, a branch of mathematics, the cyclic surgery theorem states that, for a compact, connected, orientable, irreducible three-manifold M whose boundary is a torus T, if M is

Bracket polynomial

In the mathematical field of knot theory, the bracket polynomial (also known as the Kauffman bracket) is a polynomial invariant of framed links. Although it is not an invariant of knots or links (as i

Loop representation in gauge theories and quantum gravity

Attempts have been made to describe gauge theories in terms of extended objects such as Wilson loops and holonomies. The loop representation is a quantum hamiltonian representation of gauge theories i

Tait conjectures

The Tait conjectures are three conjectures made by 19th-century mathematician Peter Guthrie Tait in his study of knots. The Tait conjectures involve concepts in knot theory such as alternating knots,

Temperley–Lieb algebra

In statistical mechanics, the Temperley–Lieb algebra is an algebra from which are built certain transfer matrices, invented by Neville Temperley and Elliott Lieb. It is also related to integrable mode

Willerton's fish

In knot theory, Willerton's fish is an unexplained relationship between the first two Vassiliev invariants of a knot. These invariants are c2, the quadratic coefficient of the Alexander–Conway polynom

String link

No description available.

Gordon–Luecke theorem

In mathematics, the Gordon–Luecke theorem on knot complements states that if the complements of two tame knots are homeomorphic, then the knots are equivalent. In particular, any homeomorphism between

Petal projection

In knot theory, a petal projection of a knot is a knot diagram with a single crossing, at which an odd number of non-nested arcs ("petals") all meet. Because the above-below relation between the branc

List of knot theory topics

Knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it

Connected sum

In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This

Regular isotopy

In the mathematical subject of knot theory, regular isotopy is the equivalence relation of link diagrams that is generated by using the 2nd and 3rd Reidemeister moves only. The notion of regular isoto

Wirtinger presentation

In mathematics, especially in group theory, a Wirtinger presentation is a finite presentation where the relations are of the form where is a word in the generators, Wilhelm Wirtinger observed that the

Braid group

In mathematics, the braid group on n strands (denoted ), also known as the Artin braid group, is the group whose elements are equivalence classes of n-braids (e.g. under ambient isotopy), and whose gr

Arithmetic topology

Arithmetic topology is an area of mathematics that is a combination of algebraic number theory and topology. It establishes an analogy between number fields and closed, orientable 3-manifolds.

Dowker–Thistlethwaite notation

In the mathematical field of knot theory, the Dowker–Thistlethwaite (DT) notation or code, for a knot is a sequence of even integers. The notation is named after Clifford Hugh Dowker and Morwen Thistl

Clasper (mathematics)

In the mathematical field of low-dimensional topology, a clasper is a surface (with extra structure) in a 3-manifold on which surgery can be performed.

Milnor conjecture (topology)

In knot theory, the Milnor conjecture says that the slice genus of the torus knot is It is in a similar vein to the Thom conjecture. It was first proved by gauge theoretic methods by Peter Kronheimer

Skein relation

Skein relations are a mathematical tool used to study knots. A central question in the mathematical theory of knots is whether two knot diagrams represent the same knot. One way to answer the question

Tangle (mathematics)

In mathematics, a tangle is generally one of two related concepts:
* In John Conway's definition, an n-tangle is a proper embedding of the disjoint union of n arcs into a 3-ball; the embedding must s

Volume conjecture

In the branch of mathematics called knot theory, the volume conjecture is the following open problem that relates quantum invariants of knots to the hyperbolic geometry of knot complements. Let O deno

Conway algebra

In mathematics, a Conway algebra, introduced by Paweł Traczyk and Józef H. Przytycki and named after John Horton Conway, is an algebraic structure with two binary operations | and * and an infinite nu

Alexander matrix

In mathematics, an Alexander matrix is a presentation matrix for the Alexander invariant of a knot. The determinant of an Alexander matrix is the Alexander polynomial for the knot.

History of knot theory

Knots have been used for basic purposes such as recording information, fastening and tying objects together, for thousands of years. The early, significant stimulus in knot theory would arrive later w

Quantum invariant

In the mathematical field of knot theory, a quantum knot invariant or quantum invariant of a knot or link is a linear sum of colored Jones polynomial of surgery presentations of the knot complement.

Gauss notation

Gauss notation (also known as a Gauss code or Gauss word) is a notation for mathematical knots. It is created by enumerating and classifying the crossings of an embedding of the knot in a plane. It is

The Knot Atlas

The Knot Atlas is a website, an encyclopedia rather than atlas, dedicated to knot theory. It and its predecessor were created by mathematician Dror Bar-Natan, who maintains the current site with Scott

Peripheral subgroup

In algebraic topology, a peripheral subgroup for a space-subspace pair X ⊃ Y is a certain subgroup of the fundamental group of the complementary space, π1(X − Y). Its conjugacy class is an invariant o

Band sum

In geometric topology, a band sum of two n-dimensional knots K1 and K2 along an (n + 1)-dimensional 1-handle h called a band is an n-dimensional knot K such that:
* There is an (n + 1)-dimensional 1-

List of prime knots

In knot theory, prime knots are those knots that are indecomposable under the operation of knot sum. The prime knots with ten or fewer crossings are listed here for quick comparison of their propertie

Thurston–Bennequin number

In the mathematical theory of knots, the Thurston–Bennequin number, or Bennequin number, of a front diagram of a Legendrian knot is defined as the writhe of the diagram minus the number of right cusps

Unknotting problem

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 algori

Fox n-coloring

In the mathematical field of knot theory, Fox n-coloring is a method of specifying a representation of a knot group or a group of a link (not to be confused with a link group) onto the dihedral group

Ménage problem

In combinatorial mathematics, the ménage problem or problème des ménages asks for the number of different ways in which it is possible to seat a set of male-female couples at a round dining table so t

Braid theory

No description available.

Planar algebra

In mathematics, planar algebras first appeared in the work of Vaughan Jones on the standard invariant of a II1 subfactor. They also provide an appropriate algebraic framework for many knot invariants

Knot thickness

In knot theory, each link and knot can have an assigned knot thickness. Each realization of a link or knot has a thickness assigned to it. The thickness τ of a link allows us to introduce a scale with

Free loop

In the mathematical field of topology, a free loop is a variant of the mathematical notion of a loop. Whereas a loop has a distinguished point on it, called a basepoint, a free loop lacks such a disti

Racks and quandles

In mathematics, racks and quandles are sets with binary operations satisfying axioms analogous to the Reidemeister moves used to manipulate knot diagrams. While mainly used to obtain invariants of kno

Seifert surface

In mathematics, a Seifert surface (named after German mathematician Herbert Seifert) is an orientable surface whose boundary is a given knot or link. Such surfaces can be used to study the properties

Knot complement

In mathematics, the knot complement of a tame knot K is the space where the knot is not. If a knot is embedded in the 3-sphere, then the complement is the 3-sphere minus the space near the knot. To ma

Milnor map

In mathematics, Milnor maps are named in honor of John Milnor, who introduced them to topology and algebraic geometry in his book Singular Points of Complex Hypersurfaces (Princeton University Press,

Signature of a knot

The signature of a knot is a topological invariant in knot theory. It may be computed from the Seifert surface. Given a knot K in the 3-sphere, it has a Seifert surface S whose boundary is K. The of S

Average crossing number

In the mathematical subject of knot theory, the average crossing number of a knot is the result of averaging over all directions the number of crossings in a knot diagram of the knot obtained by proje

Knot energy

In physical knot theory, a knot energy is a functional on the space of all knot conformations. A conformation of a knot is a particular embedding of a circle into three-dimensional space. Depending on

Knot theory

In the mathematical field of topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot diff

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