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 formula_1 be Kashaev's invariant of formula_2; this invariant coincides with the following evaluation of the formula_3-Colored Jones Polynomial formula_4 of formula_2: Then the volume conjecture states that where vol("K") 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 formula_6 of the complement of knots formula_7 and showed that it is true for the knots formula_8, formula_9, and formula_10. He conjectured that for the general hyperbolic knots the formula (2) would hold. His invariant for a knot formula_2 is based on the theory of quantum dilogarithms at the formula_3-th root of unity, formula_13. Colored Jones Invariant. had firstly pointed out that Kashaev's invariant is related to the colored Jones polynomial by replacing q with the 2N-root of unity, namely, formula_14. 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 (1) into where formula_15 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.