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(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 of the complement of knots and showed that it is true for the knots , , and . He conjectured that for the general hyperbolic knots the formula (2) 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 the colored 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: Assuming the volume conjecture, every knot that is different from the trivial knot has at least one different Vassiliev (finite type) invariant. Relation to Chern-Simons theory Using complexification, rewrote the formula (1) 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. References . . . . Knot theory Conjectures Unsolved problems in geometry