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

TL;DR

This paper investigates the asymptotic growth of the colored Jones polynomial for a cable of the figure-eight knot, revealing exponential growth linked to the Chern-Simons invariant when evaluated at specific roots of unity.

Contribution

It provides new insights into the asymptotic behavior of colored Jones polynomials for cable knots and connects growth rates to geometric invariants.

Findings

Colored Jones polynomial grows exponentially for large N when evaluated at exp(ξ/N) with large ξ.

Growth rate of the polynomial is related to the Chern-Simons invariant of the knot exterior.

Establishes a link between quantum invariants and geometric structures of the knot complement.

Abstract

We study the asymptotic behavior of the $N$-dimensional colored Jones polynomial of a cable of the figure-eight knot, evaluated at $\exp(\xi/N)$ for a real number $\xi$. We show that if $\xi$ is sufficiently large, the colored Jones polynomial grows exponentially when $N$ goes to the infinity. Moreover the growth rate is related to the Chern-Simons invariant of the knot exterior associated with an $\mathrm{SL}(2;\mathbb{R})$ representation.