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Polynomials arising in knot theory
Alexander polynomial and the Jones polynomial, both of which can be obtained by appropriate substitutions from HOMFLY. The HOMFLY polynomial is also a
HOMFLY_polynomial
Two-variable polynomial knot invariant
to Chern–Simons gauge theories for SO(N) in the same way that the HOMFLY polynomial is related to Chern–Simons gauge theories for SU(N). Kauffman, Louis
Kauffman_polynomial
Knot invariant
still unknown whether the Jones polynomial (or the stronger HOMFLY polynomial) determines the unknot. The Alexander polynomial does not detect primeness of
Alexander_polynomial
Mathematical invariant of a knot or link
crossings. HOMFLY polynomial Alexander polynomial Volume conjecture Chern–Simons theory Quantum group Jones, Vaughan F.R. (1985). "A polynomial invariant
Jones_polynomial
Jones discovered the Jones polynomial. This led to the discovery of more knot polynomials, such as the so-called HOMFLY polynomial. Soon after Jones' discovery
Knot_polynomial
Topological quantum field theory
/(k+N))}{\sin(\pi N/(k+N))}}} times the HOMFLY polynomial. In particular when N = 2 the HOMFLY polynomial reduces to the Jones polynomial. In the SO(N) case, one finds
Chern–Simons_theory
Simplest non-trivial closed knot with three crossings
{\displaystyle L(a,z)=za^{5}+z^{2}a^{4}-a^{4}+za^{3}+z^{2}a^{2}-2a^{2}.} The HOMFLY polynomial of the trefoil is L ( α , z ) = − α 4 + α 2 z 2 + 2 α 2 . {\displaystyle
Trefoil_knot
polynomials Touchard polynomials Wilkinson's polynomial Wilson polynomials Zernike polynomials Pseudo-Zernike polynomials Alexander polynomial HOMFLY
List_of_polynomial_topics
British mathematician (born 1938)
include topology and knot theory. He was one of the discoverers of the HOMFLY polynomial invariant of links, and proved the Lickorish-Wallace theorem which
W._B._R._Lickorish
Algorithm to be run on quantum computers
algorithms for estimating quantum topological invariants such as Jones and HOMFLY polynomials, and the Turaev-Viro invariant of three-dimensional manifolds. In
Quantum_algorithm
the Jones polynomial in 1984. This led to other knot polynomials such as the bracket polynomial, HOMFLY polynomial, and Kauffman polynomial. Jones was
History_of_knot_theory
Study of mathematical knots
establishes a relationship between this bracket operation and the HOMFLY-PT polynomial P {\displaystyle {\mathcal {P}}} , expressing P ( [ α , β ] ) {\displaystyle
Knot_theory
Invariant of mathematical knots
Jones polynomial P 2 ( L ) {\displaystyle P_{2}(L)} is the Euler characteristic of a bigraded link homology theory. The entire HOMFLY-PT polynomial is the
Khovanov_homology
Jones polynomial. Also known as the Kauffman bracket. Conway polynomial uses Skein relations. Homfly polynomial or HOMFLYPT polynomial. Jones polynomial assigns
List_of_knot_theory_topics
Notation for mathematical knots
Meunier-Guttin-Cluzel, S.; Letellier, C. (1999). "Computer evaluation of Homfly polynomials by using Gauss codes, with a skein-template algorithm". Applied Mathematics
Gauss_notation
Polynomial invariant of framed links
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
Bracket_polynomial
Concept in mathematical knot theory
Casson invariant Casson-Walker invariant Khovanov–Rozansky invariant HOMFLY polynomial K-theory invariants Atiyah–Patodi–Singer eta invariant Link invariant
Quantum_invariant
Database and web server for analyzing knots in protein structures
constructed using the centre of mass. Knot identification uses the HOMFLY polynomial to distinguish knot types. AlphaKnot recognizes knots with minimal
AlphaKnot
American mathematician (born 1936)
embedding theorem. In addition, Freyd's name is associated with the HOMFLYPT polynomial of knot theory, and he and Andre Scedrov originated the concept of (mathematical)
Peter_J._Freyd
Knot that is not equivalent to its mirror image
the converse is not true. The HOMFLY polynomial is even better at detecting chirality, but there is no known polynomial knot invariant that can fully
Chiral_knot
Kind of operation in knot theory
same hyperbolic volume (by a result of Ruberman), and have the same HOMFLY polynomials. Conway and Kinoshita-Terasaka mutant pair, distinguished as knot
Mutation_(knot_theory)
Mathematical tool for studying knots
Their Polynomials, Feature Column. Weisstein, Eric W. "Skein Relationship". MathWorld. Morton, Hugh R.; Lukac, Sascha G. (2003), "HOMFLY polynomial of decorated
Skein_relation
Family of mathematical knots
depend on the number n {\displaystyle n} of half-twists. The Alexander polynomial of a twist knot is given by the formula Δ ( t ) = { n + 1 2 t − n + n
Twist_knot
Mathematical knot with crossing number 6
\,} The Alexander polynomial and Conway polynomial are the same as those for the knot 946, but the Jones polynomials for these two knots are different
Stevedore_knot_(mathematics)
Link of three loops with ten crossings
{\displaystyle w(t)} is essentially the Jones polynomial for the Whitehead link.) The HOMFLY polynomial is P ( α , z ) = z − 2 α − 2 − 4 z 2 α − 2 − 4
L10a140_link
American mathematician (born 1941)
theory to DNA structure; his initial is the "M" in the name of the HOMFLY polynomial. Millett graduated from the Massachusetts Institute of Technology
Kenneth_Millett
Prime knot named for John Horton Conway
shares the same Jones polynomial. Both knots also have the property of having the same Alexander polynomial and Conway polynomial as the unknot. The issue
Conway_knot
Type of mathematical knot
Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability
Satellite_knot
Embedding of the circle in three dimensional Euclidean space
establishes a relationship between this bracket operation and the HOMFLY-PT polynomial P {\displaystyle {\mathcal {P}}} , expressing P ( [ α , β ] ) {\displaystyle
Knot_(mathematics)
Mathematical knot with crossing number 5
because of its Conway polynomial, which is ∇ ( z ) = 2 z 2 + 1 , {\displaystyle \nabla (z)=2z^{2}+1,\,} and its Jones polynomial is V ( q ) = q − 1 − q
Three-twist_knot
Knot invariant named after Cahit Arf
(t)=c_{0}+c_{1}t+\cdots +c_{n}t^{n}+\cdots +c_{0}t^{2n}} be the Alexander polynomial of the knot. Then the Arf invariant is the residue of c n − 1 + c n −
Arf_invariant_of_a_knot
Two interlinked loops with five structural crossings
matrix, or because of its Conway polynomial, which is ∇ ( z ) = z 3 . {\displaystyle \nabla (z)=z^{3}.} Its Jones polynomial is V ( t ) = t − 3 2 ( − 1 +
Whitehead_link
Mathematical knot with crossing number 7
knot. Its Alexander polynomial is Δ ( t ) = 3 t − 5 + 3 t − 1 , {\displaystyle \Delta (t)=3t-5+3t^{-1},\,} its Conway polynomial is ∇ ( z ) = 3 z 2 +
7_2_knot
Unique knot with a crossing number of four
because of its Conway polynomial, which is ∇ ( z ) = 1 − z 2 , {\displaystyle \nabla (z)=1-z^{2},\ } and the Jones polynomial is V ( q ) = q 2 − q +
Figure-eight knot (mathematics)
Figure-eight_knot_(mathematics)
Mathematical knot with crossing number 6
Alexander polynomial is Δ ( t ) = − t 2 + 3 t − 3 + 3 t − 1 − t − 2 , {\displaystyle \Delta (t)=-t^{2}+3t-3+3t^{-1}-t^{-2},\,} its Conway polynomial is ∇ (
62_knot
Mathematical knot with crossing number 5
because of its Conway polynomial, which is ∇ ( z ) = z 4 + 3 z 2 + 1 {\displaystyle \nabla (z)=z^{4}+3z^{2}+1} , and its Jones polynomial is V ( q ) = q −
Cinquefoil_knot
Determining whether a knot is the unknot
Unsolved problem in mathematics Can unknots be recognized in polynomial time? More unsolved problems in mathematics In mathematics, the unknotting problem
Unknotting_problem
Polish American mathematician
published a paper that included a description of what is now called the HOMFLY(PT) polynomial. Postal delays prevented Przytycki and Traczyk from receiving full
Józef_Przytycki
How many times curves wind around each other
Witten that the nonabelian theory gives the invariant known as the Jones polynomial. The Chern-Simons gauge theory lives in 3 spacetime dimensions. More generally
Linking_number
Loop seen as a trivial knot
through the calculation of knot invariants. The Alexander–Conway polynomial and Jones polynomial of the unknot are trivial: Δ ( t ) = 1 , ∇ ( z ) = 1 , V (
Unknot
Group whose operation is a composition of braids
theorem, was published in 1997. Vaughan Jones originally defined his polynomial as a braid invariant and then showed that it depended only on the class
Braid_group
One of three types of isotopy-preserving local changes to a knot diagram
important invariants can be defined in this way, including the Jones polynomial. The type I move is the only move that affects the writhe of the diagram
Reidemeister_move
Connected sum of two trefoil knots with same chirality
the granny knot is not a ribbon knot or a slice knot. The Alexander polynomial of the granny knot is Δ ( t ) = ( t − 1 + t − 1 ) 2 , {\displaystyle \Delta
Granny_knot_(mathematics)
Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability
Tunnel_number
Non-trivial knot which cannot be written as the knot sum of two non-trivial knots
Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability
Prime_knot
Mathematical knot with crossing number 7
Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability
74_knot
Type of knot in knot theory
Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability
2-bridge_knot
Prime knot with crossing number 10
Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability
Perko_pair
Mathematical knot with crossing number 6
Alexander polynomial of the 63 knot is Δ ( t ) = t 2 − 3 t + 5 − 3 t − 1 + t − 2 , {\displaystyle \Delta (t)=t^{2}-3t+5-3t^{-1}+t^{-2},\,} Conway polynomial is
63_knot
Property in knot theory
Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability
Tricolorability
Generalization of knots in 3-dimensional Euclidean space
problem in mathematics [Extension of Jones polynomial to general 3-manifolds.] Can the original Jones polynomial, which is defined for 1-links in the 3-sphere
Virtual_knot
Three linked but pairwise separated rings
Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability
Borromean_rings
Function of a knot that takes the same value for equivalent knots
particularly simple and common example. Other examples are knot polynomials, such as the Jones polynomial, which are currently among the most useful invariants
Knot_invariant
Knot which lies on the surface of a torus in 3-dimensional space
( q − 1 ) . {\displaystyle g={\frac {1}{2}}(p-1)(q-1).} The Alexander polynomial of a torus knot is t k ( t p q − 1 ) ( t − 1 ) ( t p − 1 ) ( t q − 1 )
Torus_knot
Mathematical knot with crossing number 7
its Conway polynomial is ∇ ( z ) = z 6 + 5 z 4 + 6 z 2 + 1 , {\displaystyle \nabla (z)=z^{6}+5z^{4}+6z^{2}+1,\,} and its Jones polynomial is V ( q ) =
71_knot
Murasugi (村杉 邦男), and Morwen Thistlethwaite in 1987, using the Jones polynomial. A second conjecture of Tait: An amphicheiral (or acheiral) alternating
Tait_conjectures
Orientable surface whose boundary is a knot or link
\left(V-tV^{*}\right),} which is a polynomial of degree at most 2g in the indeterminate t . {\displaystyle t.} The Alexander polynomial is independent of the choice
Seifert_surface
Invariant of framed knots
Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability
Self-linking_number
Connected sum of two trefoil knots with opposite chirality
smallest possible crossing number for a composite knot. The Alexander polynomial of the square knot is Δ ( t ) = ( t − 1 + t − 1 ) 2 , {\displaystyle \Delta
Square_knot_(mathematics)
Complement of a knot in three-sphere
Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability
Knot_complement
Every knot or link can be represented as a closed braid
Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability
Alexander's_theorem
Smallest number of edges of an equivalent polygonal path for a knot
Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability
Stick_number
Type of mathematical link
Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability
Hyperbolic_link
Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability
Crosscap_number
Operation on a knot
Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability
Flype
Knot that bounds an embedded disk in 4-space
Alexander polynomial of a slice knot can be written as Δ ( t ) = f ( t ) f ( t − 1 ) {\displaystyle \Delta (t)=f(t)f(t^{-1})} with a Laurent polynomial f {\displaystyle
Slice_knot
Type of mathematical knot
Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability
Ribbon_knot
Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability
Bridge_number
Mathematical notation for describing the structure of knots
Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability
Dowker–Thistlethwaite notation
Dowker–Thistlethwaite_notation
Invariant of a knot diagram
Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability
Writhe
Encyclopedic website dedicated to knot theory
Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability
The_Knot_Atlas
Motif with two doubly-interlinked loops
Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability
Solomon's_knot
Integer-valued knot invariant; least number of crossings in a knot diagram
Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability
Crossing_number_(knot_theory)
Fundamental group of a knot complement
Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability
Knot_group
Collection of knots that do not intersect, but may be linked
Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability
Link_(knot_theory)
Attempt to classify and tabulate all possible knots
Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability
Knot_tabulation
Class of mathematical knot with special properties
Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability
Berge_knot
Link formed from a finite number of twisted sections
Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability
Pretzel_link
Notation used to describe knots based on operations on tangles
Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability
Conway_notation_(knot_theory)
Type of mathematical knot
Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability
(−2,3,7)_pretzel_knot
Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability
Invertible_knot
Simplest nontrivial knot link
Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability
Hopf_link
Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability
Alternating_knot
Interlinked multi-loop construction where cutting one loop frees all the others
Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability
Brunnian_link
Link that consists of finitely many unlinked unknots
Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability
Unlink
Knot that can't be tied in a string of constant diameter
Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability
Wild_knot
Mathematical knot
Examples of knots with nonmonic Alexander polynomials abound, for example the twist knots have Alexander polynomials q t − ( 2 q + 1 ) + q t − 1 {\displaystyle
Fibered_knot
Concept in knot theory
Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability
Conway_sphere
Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability
Knot_operation
Type of invariant in Knot theory
given by the coefficient of the quadratic term of the Alexander–Conway polynomial. It is an invariant of order two. Modulo two, it is equal to the Arf invariant
Finite_type_invariant
Normalized hyperbolic volume of the complement of a hyperbolic knot
Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability
Hyperbolic_volume
Minimum number of times a specific knot must be passed through itself to become untied
Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability
Unknotting_number
Analog of the knot group
Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability
Link_group
Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability
List of mathematical knots and links
List_of_mathematical_knots_and_links
Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability
List_of_prime_knots
Flat woven decorative knot
Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability
Carrick_mat
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HOMFLY POLYNOMIAL
HOMFLY POLYNOMIAL
HOMFLY POLYNOMIAL
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