Jones Polynomial
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Jones Polynomial

In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984.[1] Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynomial in the variable ${\displaystyle t^{1/2}}$ with integer coefficients.[2]

Definition by the bracket

Type I Reidemeister move

Suppose we have an oriented link ${\displaystyle L}$, given as a knot diagram. We will define the Jones polynomial, ${\displaystyle V(L)}$, using Kauffman's bracket polynomial, which we denote by ${\displaystyle \langle ~\rangle }$. Note that here the bracket polynomial is a Laurent polynomial in the variable ${\displaystyle A}$ with integer coefficients.

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

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

where ${\displaystyle w(L)}$ denotes the writhe 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 three Reidemeister 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 multiplication by ${\displaystyle -A^{\pm 3}}$ 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 +1 or -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 ${\displaystyle t^{1/2}}$.

Jones polynomial for tangles

This construction of the Jones polynomial for tangles is a simple generalization of the Kauffman bracket of a link. The construction was developed by Professor Vladimir G. Turaev and published in 1990 in the Journal of Mathematics and Science.[3]

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 the ring of Laurent polynomials with integer coefficients in the variable ${\displaystyle t^{1/2}}$.

Definition by braid representation

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. the Potts model, in statistical mechanics.

Let a link L be given. A theorem of Alexander's states that it is the trace closure of a braid, say with n strands. Now define a representation ${\displaystyle \rho }$ of the braid group on n strands, Bn, into the Temperley-Lieb algebra TLn 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 L and compute ${\displaystyle \delta ^{n-1}tr\rho (\sigma )}$ where tr is the Markov trace. This gives ${\displaystyle \langle L\rangle }$, where ${\displaystyle \langle }$ ${\displaystyle \rangle }$ is the bracket polynomial. This can be seen by considering, as 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 the R-matrix representations, leading to "generalized Jones invariants".

Properties

The Jones polynomial is characterized by taking the value 1 on any diagram of the unknot and satisfies the following skein 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, an amphichiral knot, a knot equivalent to its mirror image, has palindromic entries in its Jones polynomial. See the article on skein 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 an alternating polynomial. This property was proved by Morwen Thistlethwaite [4] in 1987. Another proof of this last property is due to Hernando Burgos-Soto, who also gave an extension to tangles[5] of the property.

Colored Jones polynomial

N colored Jones Polynomial: The N cables of L are parallel with each other along the knot L and each is colored a different color.

For a positive integer N a N-colored Jones polynomial ${\displaystyle V_{N}(L,t)}$ can be defined as the Jones polynomial for N cables of the knot ${\displaystyle L}$ as depicted in the right figure. It is associated with an (N + 1)-dimensional irreducible representation of ${\displaystyle SU(2)}$ . The label N stands for coloring. Like the ordinary Jones polynomial it can be defined by Skein relation and is a Laurent polynomial in one variable t . The N-colored Jones polynomial ${\displaystyle V_{N}(L,t)}$ has the following properties:

• ${\displaystyle V_{X\oplus Y}(L,t)=V_{X}(L,t)+V_{Y}(L,t)}$ where ${\displaystyle X,Y}$ are two representation space.
• ${\displaystyle V_{X\otimes Y}(L,t)}$ equals the Jones polynomial of the 2-cables of L with two components labeled by ${\displaystyle X}$ and ${\displaystyle Y}$ . So the N-colored Jones polynomial equals the original Jones polynomial of the N cables of ${\displaystyle L}$ .
• The original Jones polynomial appears as a special case: ${\displaystyle V(L,t)=V_{1}(L,t)}$ .

Relationship to other theories

As first shown by Edward Witten, the Jones polynomial of a given knot ${\displaystyle \gamma }$ can be obtained by considering Chern-Simons theory on the three-sphere with gauge group SU(2), and computing the vacuum expectation value of a Wilson loop ${\displaystyle W_{F}(\gamma )}$, associated to ${\displaystyle \gamma }$, and the fundamental representation ${\displaystyle F}$ of ${\displaystyle \mathrm {SU} (2)}$.

By substituting ${\displaystyle e^{h}}$ the variable ${\displaystyle t}$ of the Jones polynomial and expanding it as the series of h each of the coefficients turn to be the Vassiliev invariant of the knot K. In order to unify the Vassiliev invariants(finite type invariant) Maxim Kontsevich constructed the Kontsevich integral. The value of the Kontsevich integral, which is the infinite sum of 1, 3-valued chord diagram, named the Jacobi chord diagram, reproduces the Jones polynomial along with the ${\displaystyle sl_{2}}$ weight system which was deeply studied by Dror Bar-Natan.

By numerical examinations on some hyperbolic knots R. M. Kashaev discovered that substituting the n-th root of unity the parameter of the colored Jones polynomial corresponding to the N-dimensional representation and limiting it as N grows to the infinity its limit value would give the hyperbolic volume of the knot complement. (See Volume conjecture.)

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

Open problems

• Is there a nontrivial knot with Jones polynomial equal to that of the unknot? It is known that there are nontrivial links with Jones polynomial equal to that of the corresponding unlinks by the work of Morwen Thistlethwaite.