Volume Conjecture and Asymptotic Expansion ofq-Series

Abstract

We consider the “volume conjecture,” which states that an asymptotic limit of Kashaev's invariant (or, the colored Jonestype invariant) of knot κ gives the hyperbolic volume of the complement of knot κ. 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).