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: , , where denotes the 2-cabling of . 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. Relationship to other theories Link with Chern–Simons theory As first shown by Edward Witten, the Jones polynomial of a given knot can be obtained by considering Chern–Simons theory on the three-sphere with gauge group , and computing the vacuum expectation value of a Wilson loop , associated to , and the fundamental representation of . Link with quantum knot invariants 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. 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