First, we define the auxiliary polynomial (also known as the normalized bracket polynomial)
where denotes the writhe of in its given diagram. The writhe of a diagram is the number of positive crossings ( in the figure below) minus the number of negative crossings (). The writhe is not a knot invariant.
is a knot invariant since it is invariant under changes of the diagram of 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 under a type I Reidemeister move. The definition of the polynomial given above is designed to nullify this change, since the writhe changes appropriately by or under type I moves.
Now make the substitution in to get the Jones polynomial . This results in a Laurent polynomial with integer coefficients in the variable .
Let be a non-negative integer and denote the set of all isotopic types of tangle diagrams, with 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 -end oriented tangle an element of the free -module , where is the ring of Laurent polynomials with integer coefficients in the variable .
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 of the braid group on n strands, Bn, into the Temperley–Lieb algebra with coefficients in and . The standard braid generator is sent to , where are the standard generators of the Temperley–Lieb algebra. It can be checked easily that this defines a representation.
Take the braid word obtained previously from and compute where is the Markov trace. This gives , where 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".
The Jones polynomial is characterized by taking the value 1 on any diagram of the unknot and satisfies the following skein relation:
where , , and 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 , the Jones polynomial of its mirror image is given by substitution of for in . 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 , the -colored Jones polynomial is a generalisation of the Jones polynomial. It is the Reshetikhin–Turaev invariant associated with the -irreducible representation of the quantum group. 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 . One thinks of the strands of a link as being "colored" by a representation, hence the name.
More generally, given a link of components and representations of , the -colored Jones polynomial is the Reshetikhin–Turaev invariant associated to (here we assume the components are ordered). Given two representations and , colored Jones polynomials satisfy the following two properties:
These properties are deduced from the fact that colored Jones polynomials are Reshetikhin-Turaev invariants.
Let be a knot. Recall that by viewing a diagram of as an element of the Temperley-Lieb algebra thanks to the Kauffman bracket, one recovers the Jones polynomial of . Similarly, the -colored Jones polynomial of can be given a combinatorial description using the Jones-Wenzl idempotents, as follows:
consider the -cabling of ;
view it as an element of the Temperley-Lieb algebra;
insert the Jones-Wenzl idempotents on some parallel strands.
The resulting element of is the -colored Jones polynomial. See appendix H of  for further details.
By substituting for the variable 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 . 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 weight system studied by Dror Bar-Natan.
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.
Problem（Extension of Jones polynomial to general 3-manifolds）
``The original Jones polynomial was defined for 1-links in the 3-sphere (the 3-ball, the 3-space ). 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 ). This problem is also open at the physics level. In the case of the Alexander polynomial, this problem is solved.
^Kauffman, Louis H.; Ogasa, Eiji; Schneider, Jonathan (2018), A spinning construction for virtual 1-knots and 2-knots, and the fiberwise and welded equivalence of virtual 1-knots, arXiv:1808.03023
Kauffman, L.E. (1998), Talks at MSRI Meeting in January 1997, AMS Meeting at University of Maryland, College Park in March 1997, Isaac Newton Institute Lecture in November 1997, Knots in Hellas Meeting in Delphi, Greece in July 1998, APCTP-NANKAI Symposium on Yang-Baxter Systems, Non-Linear Models and Applications at Seoul, Korea in October 1998, Virtual knot theory, European J. Combin. 20 (1999) 663-690, arXiv:math/9811028