The Colored Jones Polynomial, the Chern–Simons Invariant, and the Reidemeister Torsion of a Twice–Iterated Torus Knot

Abstract

A generalization of the volume conjecture relates the asymptotic behavior of the colored Jones polynomial of a knot to the Chern–Simons invariant and the Reidemeister torsion of the knot complement associated with a representation of the fundamental group to the special linear group of degree two over complex numbers. If the knot is hyperbolic, the representation can be regarded as a deformation of the holonomy representation that determines the complete hyperbolic structure. In this article, we study a similar phenomenon when the knot is a twice-iterated torus knot. In this case, the asymptotic expansion of the colored Jones polynomial splits into sums, and each summand is related to the Chern–Simons invariant and the Reidemeister torsion associated with a representation.