In themathematical field ofknot theory, theJones polynomial is aknot polynomial discovered byVaughan Jones in 1984.^[1][2] Specifically, it is aninvariant of an orientedknot orlink which assigns to each oriented knot or link aLaurent polynomial in the variable${\displaystyle t^{1/2}}$ withintegercoefficients.^[3]

Suppose we have anoriented link${\displaystyle L}$, given as aknot diagram. We will define the Jones polynomial${\displaystyle V(L)}$ by usingLouis Kauffman'sbracket polynomial, which we denote by${\displaystyle ⟨⟩}$. Here the bracket polynomial is aLaurent polynomial in the variable${\displaystyle A}$ with integer coefficients.

First, we define the auxiliary polynomial (also known as the normalized bracket polynomial)

${\displaystyle X(L)=(-A^3{)}^{-w(L)}⟨L⟩,}$

where${\displaystyle w(L)}$ denotes thewrithe of${\displaystyle L}$ in its given diagram. The writhe of a diagram is the number of positive crossings (${\displaystyle L_{+}}$ in the figure below) minus the number of negative crossings (${\displaystyle L_{-}}$). The writhe is not a knot invariant.

${\displaystyle X(L)}$ is a knot invariant since it is invariant under changes of the diagram of${\displaystyle L}$ by the threeReidemeister moves. Invariance under type II and III Reidemeister moves follows from invariance of the bracket under those moves. The bracket polynomial is known to change by a factor of${\displaystyle -A^{\pm 3}}$ under a type I Reidemeister move. The definition of the${\displaystyle X}$ polynomial given above is designed to nullify this change, since the writhe changes appropriately by${\displaystyle +1}$ or${\displaystyle -1}$ under type I moves.

Now make the substitution${\displaystyle A=t^{-1/4}}$ in${\displaystyle X(L)}$ to get the Jones polynomial${\displaystyle V(L)}$. This results in a Laurent polynomial with integer coefficients in the variable${\displaystyle t^{1/2}}$.

This construction of the Jones polynomial fortangles is a simple generalization of theKauffman bracket of a link. The construction was developed byVladimir Turaev and published in 1990.^[4]

Let${\displaystyle k}$ be a non-negative integer and${\displaystyle S_k}$ denote the set of all isotopic types of tangle diagrams, with${\displaystyle 2k}$ ends, having no crossing points and no closed components (smoothings). Turaev's construction makes use of the previous construction for the Kauffman bracket and associates to each${\displaystyle 2k}$-end oriented tangle an element of the free${\displaystyle \mathrm{R}}$-module${\displaystyle \mathrm{R}[S_k]}$, where${\displaystyle \mathrm{R}}$ is thering ofLaurent polynomials with integer coefficients in the variable${\displaystyle t^{1/2}}$.

Jones' original formulation of his polynomial came from his study of operator algebras. In Jones' approach, it resulted from a kind of "trace" of a particular braid representation into an algebra which originally arose while studying certain models, e.g. thePotts model, instatistical mechanics.

Let a linkL be given. Atheorem of Alexander states that it is the trace closure of a braid, say withn strands. Now define a representation${\displaystyle \rho }$ of thebraid group onn strands,B_n, into theTemperley–Lieb algebra${\displaystyle {\mathrm{TL}}_n}$ with coefficients in${\displaystyle \mathbb{Z}[A,A^{-1}]}$ and${\displaystyle \delta =-A^2-A^{-2}}$. The standard braid generator${\displaystyle {\sigma }_i}$ is sent to${\displaystyle A\cdot e_i+A^{-1}\cdot 1}$, where${\displaystyle 1,e_1,\dots ,e_{n-1}}$ are the standard generators of the Temperley–Lieb algebra. It can be checked easily that this defines a representation.

Take the braid word${\displaystyle \sigma }$ obtained previously from${\displaystyle L}$ and compute${\displaystyle {\delta }^{n-1}\mathrm{tr}\rho (\sigma )}$ where${\displaystyle \mathrm{tr}}$ is theMarkov trace. This gives${\displaystyle ⟨L⟩}$, where${\displaystyle ⟨}$${\displaystyle ⟩}$ is the bracket polynomial. This can be seen by considering, asLouis Kauffman did, the Temperley–Lieb algebra as a particular diagram algebra.

An advantage of this approach is that one can pick similar representations into other algebras, such as theR-matrix representations, leading to "generalized Jones invariants".

The Jones polynomial is characterized by taking the value 1 on any diagram of the unknot and satisfies the followingskein relation:

${\displaystyle (t^{1/2}-t^{-1/2})V(L_0)=t^{-1}V(L_{+})-tV(L_{-})}$

where${\displaystyle L_{+}}$,${\displaystyle L_{-}}$, and${\displaystyle L_0}$ are three oriented link diagrams that are identical except in one small region where they differ by the crossing changes or smoothing shown in the figure below:

The definition of the Jones polynomial by the bracket makes it simple to show that for a knot${\displaystyle K}$, the Jones polynomial of its mirror image is given by substitution of${\displaystyle t^{-1}}$ for${\displaystyle t}$ in${\displaystyle V(K)}$. Thus, anamphicheiral knot, a knot equivalent to its mirror image, haspalindromic entries in its Jones polynomial. See the article onskein relation for an example of a computation using these relations.

Another remarkable property of this invariant states that the Jones polynomial of an alternating link is analternating polynomial. This property was proved byMorwen Thistlethwaite^[5] in 1987. Another proof of this last property is due toHernando Burgos-Soto, who also gave an extension of the property to tangles.^[6]

The Jones polynomial is not a complete invariant. There exist an infinite number of non-equivalent knots that have the same Jones polynomial. An example of two distinct knots having the same Jones polynomial can be found in the book by Murasugi.^[7]

For a positive integer${\displaystyle N}$, the${\displaystyle N}$-colored Jones polynomial${\displaystyle V_N(L,t)}$ is a generalisation of the Jones polynomial. It is theReshetikhin–Turaev invariant associated with the${\displaystyle (N+1)}$-irreducible representation of thequantum group${\displaystyle U_q({\mathfrak{s}\mathfrak{l}}_2)}$. In this scheme, the Jones polynomial is the 1-colored Jones polynomial, the Reshetikhin-Turaev invariant associated to the standard representation (irreducible and two-dimensional) of${\displaystyle U_q({\mathfrak{s}\mathfrak{l}}_2)}$. One thinks of the strands of a link as being "colored" by a representation, hence the name.

More generally, given a link${\displaystyle L}$ of${\displaystyle k}$ components and representations${\displaystyle V_1,\dots ,V_k}$ of${\displaystyle U_q({\mathfrak{s}\mathfrak{l}}_2)}$, the${\displaystyle (V_1,\dots ,V_k)}$-colored Jones polynomial${\displaystyle V_{V_1,\dots ,V_k}(L,t)}$ is theReshetikhin–Turaev invariant associated to${\displaystyle V_1,\dots ,V_k}$ (here we assume the components are ordered). Given two representations${\displaystyle V}$ and${\displaystyle W}$, colored Jones polynomials satisfy the following two properties:^[8]

• ${\displaystyle V_{V\oplus W}(L,t)=V_V(L,t)+V_W(L,t)}$,
• ${\displaystyle V_{V\otimes W}(L,t)=V_{V,W}(L^2,t)}$, where${\displaystyle L^2}$ denotes the2-cabling of${\displaystyle L}$.

These properties are deduced from the fact that colored Jones polynomials are Reshetikhin-Turaev invariants.

Let${\displaystyle K}$ be a knot. Recall that by viewing a diagram of${\displaystyle K}$ as an element of the Temperley-Lieb algebra thanks to the Kauffman bracket, one recovers the Jones polynomial of${\displaystyle K}$. Similarly, the${\displaystyle N}$-colored Jones polynomial of${\displaystyle K}$ can be given a combinatorial description using theJones-Wenzl idempotents, as follows:

• consider the${\displaystyle N}$-cabling${\displaystyle K^N}$ of${\displaystyle K}$;
• view it as an element of the Temperley-Lieb algebra;
• insert the Jones-Wenzl idempotents on some${\displaystyle N}$ parallel strands.

The resulting element of${\displaystyle \mathbb{Q}(t)}$ is the${\displaystyle N}$-colored Jones polynomial. See appendix H of^[9] for further details.

As first shown byEdward Witten,^[10] the Jones polynomial of a given knot${\displaystyle \gamma }$ can be obtained by consideringChern–Simons theory on the three-sphere withgauge group${\displaystyle \mathrm{S}\mathrm{U}(2)}$, and computing thevacuum expectation value of aWilson loop${\displaystyle W_F(\gamma )}$, associated to${\displaystyle \gamma }$, and thefundamental representation${\displaystyle F}$ of${\displaystyle \mathrm{S}\mathrm{U}(2)}$.

By substituting${\displaystyle e^h}$ for the variable${\displaystyle t}$ of the Jones polynomial and expanding it as the series of h each of the coefficients turn to be theVassiliev invariant of the knot${\displaystyle K}$. In order to unify the Vassiliev invariants (or, finite type invariants),Maxim Kontsevich constructed theKontsevich integral. The value of the Kontsevich integral, which is the infinite sum of 1, 3-valuedchord diagrams, named the Jacobi chord diagrams, reproduces the Jones polynomial along with the${\displaystyle {\mathfrak{s}\mathfrak{l}}_2}$ weight system studied byDror Bar-Natan.

By numerical examinations on some hyperbolic knots,Rinat Kashaev discovered that substituting then-th root of unity into the parameter of thecolored Jones polynomial corresponding to then-dimensional representation, and limiting it asn grows to infinity, the limit value would give thehyperbolic volume of theknot complement. (SeeVolume conjecture.)

In 2000Mikhail Khovanov constructed a certain chain complex for knots and links and showed that the homology induced from it is a knot invariant (seeKhovanov homology). The Jones polynomial is described as theEuler characteristic for this homology.

It is anopen question whether there is a nontrivial knot with Jones polynomial equal to that of theunknot. It is known that there are nontriviallinks with Jones polynomial equal to that of the correspondingunlinks by the work ofMorwen Thistlethwaite.^[11] The simplest such example has 15 essential crossings and consists of atrefoil knot linked to afigure-eight knot. It was shown by Kronheimer and Mrowka that there is no nontrivial knot with Khovanov homology equal to that of the unknot.^[12]

• Adams, Colin (2000-12-06).The Knot Book.American Mathematical Society.ISBN 0-8050-7380-9.
• Jones, Vaughan."The Jones Polynomial"(PDF).
• Jones, Vaughan (1987). "Hecke algebra representations of braid groups and link polynomials".Annals of Mathematics.126 (2):335–388.doi:10.2307/1971403.JSTOR 1971403.
• Kauffman, Louis H. (1987)."State models and the Jones polynomial".Topology.26 (3):395–407.doi:10.1016/0040-9383(87)90009-7. (explains the definition by bracket polynomial and its relation to Jones' formulation by braid representation)
• Lickorish, W. B. Raymond (1997).An introduction to knot theory. New York; Berlin; Heidelberg; Barcelona; Budapest; Hong Kong; London; Milan; Paris; Santa Clara; Singapore; Tokyo: Springer. p. 175.ISBN 978-0-387-98254-0.
• Thistlethwaite, Morwen (2001). "Links with trivial Jones polynomial".Journal of Knot Theory and Its Ramifications.10 (4):641–643.doi:10.1142/S0218216501001050.
• Eliahou, Shalom;Kauffman, Louis H.;Thistlethwaite, Morwen B. (2003)."Infinite families of links with trivial Jones polynomial".Topology.42 (1):155–169.doi:10.1016/S0040-9383(02)00012-5.
• Przytycki, Józef H. (1991). "Skein modules of 3-manifolds".Bulletin of the Polish Academy of Sciences.39 (1–2):91–100.arXiv:math/0611797.

• "Jones-Conway polynomial",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
• Links with trivial Jones polynomial byMorwen Thistlethwaite
• "The Jones Polynomial",The Knot Atlas.

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• Polynomials
• Knot invariants

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