Volume conjecture


In the branch of mathematics called knot theory, the volume conjecture is the following open problem that relates quantum invariants of knots to the hyperbolic geometry of knot complements.
Let O denote the unknot. For any knot K let be Kashaev's invariant of ; this invariant coincides with the following evaluation of the -colored Jones polynomial of :
Then the volume conjecture states that
where vol denotes the hyperbolic volume of the complement of K in the 3-sphere.

Kashaev's Observation

observed that the asymptotic behavior of a certain state sum of knots gives the hyperbolic volume of the complement of knots and showed that it is true for the knots Figure-eight knot|, Three-twist knot|, and Stevedore knot |. He conjectured that for the general hyperbolic knots the formula would hold. His invariant for a knot is based on the theory of quantum dilogarithms at the -th root of unity,.

Colored Jones Invariant

had firstly pointed out that Kashaev's invariant is related to Jones polynomial by replacing q with the 2N-root of unity, namely,. They used an R-matrix as the discrete Fourier transform for the equivalence of these two values.
The volume conjecture is important for knot theory. In section 5 of this paper they state that:

Relation to Chern-Simons theory

Using complexification, rewrote the formula into
where is called the Chern–Simons invariant. They showed that there is a clear relation between the complexified colored Jones polynomial and Chern–Simons theory from a mathematical point of view.