Hyperbolic structure arising from a knot invariant (Q2747052)

``We study the knot invariant based on the quantum dilogarithm function. This invariant can be regarded as a noncompact analog of Kashaev's invariant, or the colored Jones invariant (see the following review); it is based on an infinite dimensional representation of the quantum dilogarithm function , and both the \(R\)-matrix and the invariant are defined in an integral form. The three-dimensional picture of our invariant originates from the pentagon identity of the quantum dilogarithm function, and we show that the hyperbolicity consistency conditions in gluing hyperbolic polyhedra arise naturally in the classical limit as the saddle point equation of our invariant, thus demonstrating how the hyperbolic structure appears in the classical limit of the noncompact Jones polynomial.'' The context here is Kashaev's conjecture about the connection between his invariants and the hyperbolic volume of the complement of a knot (see the following review); a connection is suggested by the observation that the hyperbolic volume of the ideal tetrahedron is given by the Lobachevsky function which is closely related with the dilogarithm function.