Volume Conjecture and Asymptotic Expansion of q-Series

Abstract

We consider the "volume conjecture,'' which states that an asymptotic limit of Kashaev's invariant (or, the colored Jones type invariant) of knot {\small $\mathcal{K}$} gives the hyperbolic volume of the complement of knot {\small $\mathcal{K}$}. In the first part, we analytically study an asymptotic behavior of the invariant for the torus knot, and propose identities concerning an asymptotic expansion of q-series which reduces to the invariant with q being the N-th root of unity. This is a generalization of an identity recently studied by Zagier. In the second part, we show that "volume conjecture'' is numerically supported for hyperbolic knots and links (knots up to 6-crossing, Whitehead link, and Borromean rings).