List of knot theory topics - AbsoluteAstronomy.com

Knot theory
Knot theory
In topology, 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 cannot be undone. In precise mathematical language, a knot is an embedding of a...

is the study of mathematical knot

Knot (mathematics)

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...

s. 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 cannot be undone. In precise mathematical language, a knot is an embedding

Embedding

In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....

of a circle

Circle

A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

in 3-dimensional Euclidean space

Euclidean space

In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

, R³. 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

Ambient isotopy

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, taking a submanifold to another submanifold. For example in knot theory, one considers two knots the same if one can distort one knot into the...

); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.

Knots, links, braids

Knot (mathematics)
Knot (mathematics)
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...

gives a general introduction to the concept of a knot.
- Two classes of knots: torus knots
  Torus knot
  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 torus 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...
  
  and pretzel knots
- Cinquefoil knot
  Cinquefoil knot
  In knot theory, the cinquefoil knot, also known as Solomon's seal knot or the pentafoil knot, is one of two knots with crossing number five, the other being the three-twist knot. It is listed as the 51 knot in the Alexander-Briggs notation, and can also be described as the -torus knot...
  
  also known as a (5, 2) torus knot.
- Figure-eight knot (mathematics)
  Figure-eight knot (mathematics)
  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 and trefoil knot...
  
  the only 4-crossing knot
- Granny knot (mathematics)
  Granny knot (mathematics)
  In knot theory, the granny knot is a composite knot obtained by taking the connected sum of two identical trefoil knots. It is closely related to the square knot, which can also be described as a connected sum of two trefoils...
  
  and Square knot (mathematics)
  Square knot (mathematics)
  In knot theory, the square knot is a composite knot obtained by taking the connected sum of two trefoil knots. It is closely related to the granny knot, which is also a connected sum of two trefoils...
  
  are a connected sum of two Trefoil knot
  Trefoil knot
  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 loose ends of a common overhand knot, resulting in a knotted loop...
  
  s
- Perko pair
  Perko pair
  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. In Rolfsen's knot table, this supposed pair of distinct knots is labeled 10161 and 10162...
  
  , two entries in a knot table that were later shown to be identical.
- Stevedore knot (mathematics)
  Stevedore knot (mathematics)
  In knot theory, the stevedore knot is one of three prime knots with crossing number six, the others being the 62 knot and the 63 knot. The stevedore knot is listed as the 61 knot in the Alexander–Briggs notation, and it can also be described as a twist knot with four twists, or as the pretzel...
  
  , a prime knot
  Prime knot
  In knot theory, a prime knot is a knot that is, in a certain sense, indecomposable. Specifically, it is a non-trivial knot which cannot be written as the knot sum of two non-trivial knots. Knots that are not prime are said to be composite. It can be a nontrivial problem to determine whether a...
  
  with crossing number 6
- Solomon's knot
  Solomon's knot
  Solomon's knot is the most common name for a traditional decorative motif used since ancient times, and found in many cultures...
- Three-twist knot
  Three-twist knot
  In knot theory, the three-twist knot is the twist knot with three-half twists. It is listed as the 52 knot in the Alexander-Briggs notation, and is one of two knots with crossing number five, the other being the cinquefoil knot....
  
  is the twist knot with three-half twists, also known as the 5₂ knot.
- Trefoil knot
  Trefoil knot
  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 loose ends of a common overhand knot, resulting in a knotted loop...
  
  A knot with crossing number 3
- Unknot
  Unknot
  The unknot arises in the mathematical theory of knots. Intuitively, the unknot is a closed loop of rope without a knot in it. A knot theorist would describe the unknot as an image of any embedding that can be deformed, i.e. ambient-isotoped, to the standard unknot, i.e. the embedding of the...

Knot complement
Knot complement
In mathematics, the knot complement of a tame knot K is the complement of the interior of the embedding of a solid torus into the 3-sphere. To make this precise, suppose that K is a knot in a three-manifold M. Let N be a thickened neighborhood of K; so N is a solid torus...

, a compact 3 manifold obtained by removing an open neighborhood of a proper embedding of a tame knot from the 3-sphere.
Knots and graphs
Knots and graphs
Knots and graph theory are related in some simple ways.- Knot diagram :A knot in R3 , can be projected onto a plane R2 Knots and graph theory are related in some simple ways.- Knot diagram :A knot in R3 (respectively in the 3-sphere, S3), can be projected onto a plane R2 Knots and...

general introduction to knots with mention of Reidemeister move
Reidemeister move
In the mathematical area of knot theory, a Reidemeister move refers to one of three local moves on a link diagram. In 1926, Kurt Reidemeister and independently, in 1927, J.W. Alexander and G.B...

s

Notation used in knot theory:

Conway notation (knot theory)
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 construct it.-Tangles:...
Dowker notation
Dowker notation
thumb|200px|A knot diagram with crossings labelled for a Dowker sequenceIn 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...

General knot types

2-bridge knot
2-bridge knot
In the mathematical field of knot theory, a 2-bridge knot is a knot which can be isotoped so that the natural height function given by the z-coordinate has only two maxima and two minima as critical points....
Alternating knot
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. A link is alternating if it has an alternating diagram....

; a knot that can be represented by an aletrnating diagram (ie the crossing alternate over and under as one traverses the knot).
Berge knot
Berge knot
In the mathematical theory of knots, a Berge knot or doubly primitive knot is any member of a particular family of knots in the 3-sphere. A Berge knot K is defined by the conditions:# K lies on a genus two Heegaard surface S...

a class of knots related to Lens space
Lens space
A lens space is an example of a topological space, considered in mathematics. The term often refers to a specific class of 3-manifolds, but in general can be defined for higher dimensions....

surgeries
Dehn surgery
In topology, a branch of mathematics, a Dehn surgery, named after Max Dehn, is a specific construction used to modify 3-manifolds. The process takes as input a 3-manifold together with a link...

and defined in terms of their properties with respect to a genus 2 Heegaard surface.
Cable knot, see Satellite knot
Satellite knot
In the mathematical theory of knots, a satellite knot is a knot that contains an incompressible, non-boundary parallel torus in its complement. The class of satellite knots include composite knots, cable knots and Whitehead doubles. A satellite knot K can be picturesquely described as follows:...
Chiral knot is knot which is not equivalent to its mirror image.
Double torus knot
Double torus knot
A double torus knot is a closed curve drawn on the surface called a double torus . More technically, a double torus knot is the homeomorphic image of a circle in S³ which can be realized as a subset of a genus two handlebody in S³...

, a knot that can be embedded in a double torus (a genus 2 surface).
Fibered knot
Fibered knot
A knot or link Kin the 3-dimensional sphere S^3 is called fibered or fibred if there is a 1-parameter family F_t of Seifert surfaces for K, where the parameter t runs through the points of the unit circle S^1, such that if s is not equal to t...
Framed knot
Framed knot
In the mathematical theory of knots, a framed knot is the extension of a tame knot to an embedding of the solid torus D2 × S1 in S3....
Invertible knot
Invertible knot
In mathematics, especially in the area of topology known as knot theory, an invertible knot is a knot that can be continuously deformed to itself, but with its orientation reversed. A non-invertible knot is any knot which does not have this property. The invertibility of a knot is a knot invariant...
Prime knot
Prime knot
In knot theory, a prime knot is a knot that is, in a certain sense, indecomposable. Specifically, it is a non-trivial knot which cannot be written as the knot sum of two non-trivial knots. Knots that are not prime are said to be composite. It can be a nontrivial problem to determine whether a...
Legendrian knot
Legendrian knot
In mathematics, a Legendrian knot often refers to a smooth embedding of the circle into \mathbb R^3, which is tangent to the standard contact structure on \mathbb R^3...

are knots embedded in tangent to the standard contact structure.
Lissajous knot
Lissajous knot
In knot theory, a Lissajous knot is a knot defined by parametric equations of the formx = \cos,\qquad y = \cos, \qquad z = \cos,...
Ribbon knot
Ribbon knot
In the mathematical area of knot theory, a ribbon knot is a knot that bounds a self-intersecting disc with only ribbon singularities. This type of singularity is a self-intersection along an arc; the preimage of this arc consists of two arcs in the disc, one properly embedded in the disc and the...
Satellite knot
Satellite knot
In the mathematical theory of knots, a satellite knot is a knot that contains an incompressible, non-boundary parallel torus in its complement. The class of satellite knots include composite knots, cable knots and Whitehead doubles. A satellite knot K can be picturesquely described as follows:...
Slice knot
Torus knot
Torus knot
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 torus 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...
Transverse knot
Transverse knot
In mathematics, a transverse knot is a smooth embedding of a circle into a three-dimensional contact manifold such that the tangent vector at every point of the knot is transverse to the contact plane at that point....
Twist knot
Twist knot
In knot theory, a branch of mathematics, a twist knot is a knot obtained by repeatedly twisting a closed loop and then linking the ends together...
Virtual knot
Virtual knot
In knot theory, a virtual knot is a generalization of the classical idea of knots in several ways that are all equivalent, introduced by .-Overview:...
Wild knot
Wild knot
In the mathematical theory of knots, a knot is tame if it can be "thickened up", that is, if there exists an extension to an embedding of the solid torus S 1 × D 2 into the 3-sphere. A knot is tame if and only if it can be represented as a finite closed polygonal chain...

Tangles

Tangle (mathematics)
Algebraic tangle

Operations

Band sum
Band sum
In geometric topology, a band sum of two n-dimensional knots K1 and K2 along an -dimensional 1-handle h called a band is an n-dimensional knot K such that:...
Flype
Flype
In the mathematical theory of knots, a flype is a kind of manipulation of knot and link diagramsused in the Tait flyping conjecture.It consists of twisting a part of a knot, a tangle: T by 180 degrees. Flype comes from an old Scottish word meaning to fold or to turn back. Two reduced alternating...
Fox n-coloring
Fox n-coloring
In the mathematical field of knot theory, Fox n-coloring is a method of specifying a representation of a knot group onto the dihedral group of order n where n is an odd integer by coloring arcs in a link diagram...
- Tricolorability
  Tricolorability
  In the mathematical field of knot theory, the tricolorability of a knot is the ability of a knot to be colored with three colors subject to certain rules. Tricolorability is an isotopy invariant, and hence can be used to distinguish between two different knots...
Knot sum
Reidemeister move
Reidemeister move
In the mathematical area of knot theory, a Reidemeister move refers to one of three local moves on a link diagram. In 1926, Kurt Reidemeister and independently, in 1927, J.W. Alexander and G.B...

Invariants and properties

Knot invariant
Knot invariant
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 often given by ambient isotopy but can be given by homeomorphism. Some invariants are indeed numbers, but invariants can range from the...

is an invariant defined on knots which is invariant under ambient isotopies of the knot.
Finite type invariant
Finite type invariant
In the mathematical theory of knots, a finite type invariant is a knot invariant that can be extended to an invariant of certain singular knots that vanishes on singular knots with m + 1 singularities and does not vanish on some singular knot with 'm' singularities...

is a knot invariant that can be extended to an invariant of certain singular knots

Knot polynomial
Knot 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 the properties of a given knot.-History:The first knot polynomial, the Alexander polynomial, was introduced by J. W...

is a knot invariant in the form of a polynomial whose coefficients encode some of the properties of a given knot.
- Alexander polynomial
  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 polynomial, in 1923...
  
  and the associated Alexander matrix
  Alexander matrix
  In mathematics, an Alexander matrix is a presentation matrix for the Alexander invariant of a knot....
  
  ; The first knot polynomial (1923). Sometimes called the Alexander–Conway polynomial
- Bracket polynomial
  Bracket polynomial
  In the mathematical field of knot theory, the bracket polynomial is a polynomial invariant of framed links. Although it is not an invariant of knots or links , a suitably "normalized" version yields the famous knot invariant called the Jones polynomial...
  
  is a polynomial invariant of framed links. Related to the Jones polynomial. Also known as the Kauffman bracket.
- Conway polynomial
  Conway polynomial
  In mathematics, Conway polynomial can refer to:* the Alexander–Conway polynomial in knot theory* the Conway polynomial...
  
  uses Skein relation
  Skein relation
  A central question in the mathematical theory of knots is whether two knot diagrams represent the same knot. One tool used to answer such questions is a knot polynomial which is an invariant of the knot. If two diagrams have different polynomials, they represent different knots. The reverse may not...
  
  s.
- Homfly polynomial
  HOMFLY polynomial
  In the mathematical field of knot theory, the HOMFLY polynomial, sometimes called the HOMFLY-PT polynomial or the generalized Jones polynomial, is a 2-variable knot polynomial, i.e. a knot invariant in the form of a polynomial of variables m and l....
  
  or HOMFLYPT polynomial.
- Jones polynomial assigns a Laurent polynomial in the variable t^1/2 to the knot or link.
- Kauffman polynomial is a 2-variable knot polynomial due to Louis Kauffman.

Arf invariant of a knot
Average crossing number
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 projection onto the plane orthogonal to the direction...
Bridge number
Bridge number
In a mathematical field of knot theory, the bridge number is an invariant of a knot. It is defined as the minimal number of bridges required in all the possible bridge representations of a knot...
Crosscap number
Crosscap number
In the mathematical field of knot theory, the crosscap number of a knot K is the minimum of1 - \chi, \, taken over all compact, connected, non-orientable surfaces S bounding K; here \chi is the Euler characteristic...
Crossing number (knot theory)
Crossing number (knot theory)
In the mathematical area of knot theory, the crossing number of a knot is the minimal number of crossings of any diagram of the knot. It is a knot invariant....
Hyperbolic volume (knot)
Hyperbolic volume (knot)
In the mathematical field of knot theory, the hyperbolic volume of a hyperbolic link is simply the volume of the link's complement with respect to its complete hyperbolic metric. The volume is necessarily finite. The hyperbolic volume of a non-hyperbolic knot is often defined to be zero...
Kontsevich invariant
Kontsevich invariant
In the mathematical theory of knots, the Kontsevich invariant, also known as the Kontsevich integral, of an oriented framed link is the universal finite type invariant in the sense that any coefficient of the Kontsevich invariant is a finite type invariant, and any finite type invariant can be...
Linking number
Linking number
In mathematics, the linking number is a numerical invariant that describes the linking of two closed curves in three-dimensional space. Intuitively, the linking number represents the number of times that each curve winds around the other...
Milnor invariants
Racks and quandles
Racks and quandles
In mathematics, racks and quandles are sets with a binary operations satisfying axioms analogous to the Reidemeister moves used to manipulate knot diagrams....

and Biquandle
Biquandle
In mathematics, biquandles and biracks are generalizations of quandles and racks. Whereas the hinterland of quandles and racks is the theory of classical knots, that of the bi-versions, is the theory of virtual knots....
Ropelength
Ropelength
In knot theory each realization of a link or knot has an associated ropelength. Intuitively this is the minimal length of an ideally flexible rope that is needed to tie a given link, or knot...
Seifert surface
Seifert surface
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 associated knot or link. For example, many knot invariants are most easily calculated using a Seifert surface...
Self-linking number
Self-linking number
In knot theory, the self-linking number is an invariant of framed knots. It is related to the linking number of curves.A framing of a knot is a choice of a non-tangent vector at each point of the knot...
Signature of a knot
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...
Skein relation
Skein relation
A central question in the mathematical theory of knots is whether two knot diagrams represent the same knot. One tool used to answer such questions is a knot polynomial which is an invariant of the knot. If two diagrams have different polynomials, they represent different knots. The reverse may not...
Slice genus
Slice genus
In mathematics, the slice genus of a smooth knot K in S3 is the least integer g such that K is the boundary of a connected, orientable 2-manifold S of genus g embedded in the 4-ball D4 bounded by S3.More precisely, if S is required to be smoothly embedded, then this integer g is the...
Writhe
Writhe
In knot theory, the writhe is a property of an oriented link diagram. The writhe is the total number of positive crossings minus the total number of negative crossings....

Mathematical problems

Berge conjecture
Berge conjecture
In the mathematical subject of knot theory, the Berge conjecture states that the only knots in the 3-sphere which admit lens space surgeries are Berge knots. The conjecture is named after John Berge....
Birman–Wenzl algebra
Clasper (mathematics)
Clasper (mathematics)
In the mathematical field of low-dimensional topology, a clasper is a surface in a 3-manifold on which surgery can be performed.- Motivation :...
Eilenberg–Mazur swindle
Fary–Milnor theorem
Gordon–Luecke theorem
Khovanov homology
Khovanov homology
In mathematics, Khovanov homology is an invariant of oriented knots and links that arises as the homology of a chain complex. It may be regarded as a categorification of the Jones polynomial....
Knot group
Knot group
In mathematics, a knot is an embedding of a circle into 3-dimensional Euclidean space. The knot group of a knot K is defined as the fundamental group of the knot complement of K in R3,\pi_1....
Knot tabulation
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.-Beginnings:...
Knot theory
Knot theory
In topology, 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 cannot be undone. In precise mathematical language, a knot is an embedding of a...
Knotless embedding
Linkless embedding
Linkless embedding
In topological graph theory, a mathematical discipline, a linkless embedding of an undirected graph is an embedding of the graph into Euclidean space in such a way that no two cycles of the graph have nonzero linking number. A flat embedding is an embedding with the property that every cycle is the...
Link concordance
Link group
Link group
In knot theory, an area of mathematics, the link group of a link is an analog of the knot group of a knot. They were described by John Milnor in his Bachelor's thesis, .- Definition :...
Link (knot theory)
Link (knot theory)
In mathematics, 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...
Milnor conjecture (topology)
Milnor conjecture (topology)
In knot theory, the Milnor conjecture says that the slice genus of the torus knot is/2.It is in a similar vein to the Thom conjecture....
Milnor map
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 and earlier lectures. The most studied Milnor maps are actually fibrations, and the phrase Milnor fibration is more commonly...
Möbius energy
Mutation (knot theory)
Mutation (knot theory)
In the mathematical field of knot theory, a mutation is an operation on a knot that can produce different knots. Suppose K is a knot given in the form of a knot diagram. Consider a disc D in the projection plane of the diagram whose boundary circle intersects K exactly four times...
Physical knot theory
Physical knot theory
Physical knot theory is the study of mathematical models of knotting phenomena, often motivated by physical considerations from biology, chemistry, and physics. Traditional knot theory models a knot as a simple closed loop in three dimensional space. Such a knot has no thickness or physical...
Planar algebra
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 , and have been used in describing the properties of Khovanov homology with respect to tangle...
Smith conjecture
Smith conjecture
In mathematics, the Smith conjecture states that if f is a diffeomorphism of the 3-sphere, of finite order then the fixed point set of f cannot be a nontrivial knot....
Tait conjectures
Tait conjectures
The Tait conjectures are conjectures made by Peter Guthrie Tait in his study of knots. The Tait conjectures involve concepts in knot theory such as alternating knots, chirality, and writhe...
Temperley–Lieb algebra
Thurston–Bennequin number
Tricolorability
Tricolorability
In the mathematical field of knot theory, the tricolorability of a knot is the ability of a knot to be colored with three colors subject to certain rules. Tricolorability is an isotopy invariant, and hence can be used to distinguish between two different knots...
Unknotting number
Unknotting number
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...
Unknotting problem
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 algorithms...
Volume conjecture
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 denote the unknot...