The colored Jones polynomial of a cable of the figure-eight knot

Abstract

We study the asymptotic behavior of the [Formula: see text]-dimensional colored Jones polynomial of a cable of the figure-eight knot, evaluated at [Formula: see text] for a real number [Formula: see text]. We show that if [Formula: see text] is sufficiently large, the colored Jones polynomial grows exponentially when [Formula: see text] goes to the infinity. Moreover the growth rate is related to the Chern–Simons invariant of the knot exterior associated with an [Formula: see text] representation.