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 each oriented knot or link a Laurent polynomial in the variable formula_1 with integer coefficients.
Definition by the bracket.
Suppose we have an oriented link formula_2, given as a knot diagram. We will define the Jones polynomial, formula_3, using Louis Kauffman's bracket polynomial, which we denote by formula_4. Here the bracket polynomial is a Laurent polynomial in the variable formula_5 with integer coefficients.
First, we define the auxiliary polynomial (also known as the normalized bracket polynomial) 
where formula_7 denotes the writhe of formula_2 in its given diagram. The writhe of a diagram is the number of positive crossings (formula_9 in the figure below) minus the number of negative crossings (formula_10). The writhe is not a knot invariant.
formula_11 is a knot invariant since it is invariant under changes of the diagram of formula_2 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 a factor of formula_13 under a type I Reidemeister move. The definition of the formula_14 polynomial given above is designed to nullify this change, since the writhe changes appropriately by formula_15 or formula_16 under type I moves.
Now make the substitution formula_17 in formula_11 to get the Jones polynomial formula_3. This results in a Laurent polynomial with integer coefficients in the variable formula_1.
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 Vladimir Turaev and published in 1990.
Let formula_21 be a non-negative integer and formula_22 denote the set of all isotopic types of tangle diagrams, with formula_23 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 formula_23-end oriented tangle an element of the free formula_25-module formula_26, where formula_25 is the ring of Laurent polynomials with integer coefficients in the variable formula_1.
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 states that it is the trace closure of a braid, say with "n" strands. Now define a representation formula_29 of the braid group on "n" strands, "Bn", into the Temperley–Lieb algebra formula_30 with coefficients in formula_31 and formula_32. The standard braid generator formula_33 is sent to formula_34, where formula_35 are the standard generators of the Temperley–Lieb algebra. It can be checked easily that this defines a representation.
Take the braid word formula_36 obtained previously from formula_2 and compute formula_38 where formula_39 is the Markov trace. This gives formula_40, where formula_41 formula_42 is the bracket polynomial. This can be seen by considering, as Louis 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:
where formula_9, formula_10, and formula_46 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 formula_47, the Jones polynomial of its mirror image is given by substitution of formula_48 for formula_49 in formula_50. Thus, an amphicheiral 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 in 1987. Another proof of this last property is due to Hernando Burgos-Soto, who also gave an extension to tangles of the property.
Colored Jones polynomial.
For a positive integer "N" a "N"-colored Jones polynomial formula_51 can be defined as the Jones polynomial for "N" cables of the knot formula_2 as depicted in the right figure. It is associated with an formula_53-dimensional irreducible representation of formula_54 . 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 formula_51 has the following properties:
Relationship to other theories.
Link with Chern–Simons theory.
As first shown by Edward Witten, the Jones polynomial of a given knot formula_63 can be obtained by considering Chern–Simons theory on the three-sphere with gauge group formula_64, and computing the vacuum expectation value of a Wilson loop formula_65, associated to formula_63, and the fundamental representation formula_67 of formula_64.
Link with quantum knot invariants.
By substituting formula_69 the variable formula_49 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 formula_47. In order to unify the Vassiliev invariants (or, finite type invariants), Maxim Kontsevich constructed the Kontsevich integral. The value of the Kontsevich integral, which is the infinite sum of 1, 3-valued chord diagrams, named the Jacobi chord diagrams, reproduces the Jones polynomial along with the formula_72 weight system studied by Dror Bar-Natan.
Link with the volume conjecture.
By numerical examinations on some hyperbolic knots, Rinat Kashaev discovered that substituting the "n"-th root of unity into the parameter of the colored Jones polynomial corresponding to the "n"-dimensional representation, and limiting it as "n" grows to infinity, the limit value would give the hyperbolic volume of the knot complement. (See Volume conjecture.)
Link with Khovanov homology.
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.
``The original Jones polynomial was defined for 1-links in the 3-sphere (the 3-ball, the 3-space formula_73). Can you define the Jones polynomial for 1-links in any 3-manifold?’’
Such an approach was proposed by Józef H. Przytycki under the name of skein modules. In particular, the Kauffman bracket skein module and HOMFLYPT skein module. 
See section 1.1 of this paper for the background and the history of this problem. Kauffman submitted a solution in the case of the product manifold of closed oriented surface and the closed interval, by introducing virtual 1-knots. It is open in the other cases. Witten’s path integral for the Jones polynomial is written formally for links in any compact 3-manifold, but the calculus is not done even at the physics level in any other case than the 3-sphere (the 3-ball, or the 3-space formula_73). This problem is also open at the physics level. In the case of the Alexander polynomial, this problem is solved.