The colored Jones polynomial, the Chern--Simons invariant, and the Reidemeister torsion of a twice-iterated torus knot

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.