On the asymptotic behavior of the colored Jones polynomial of the figure-eight knot associated with a real number

TL;DR

This paper investigates the asymptotic properties of the colored Jones polynomial of the figure-eight knot evaluated at a specific exponential scaling, revealing connections to geometric invariants like Chern--Simons invariant and Reidemeister torsion.

Contribution

It establishes a link between the asymptotic behavior of the colored Jones polynomial and geometric invariants associated with a representation determined by a real parameter.

Findings

Asymptotic behavior encodes the SL(2,C) Chern--Simons invariant.

Reidemeister torsion can be extracted from the polynomial's asymptotics.

Results connect quantum invariants with classical geometric structures.

Abstract

We study the asymptotic behavior of the $N$-dimensional colored Jones polynomial evaluated at $\exp(\xi/N)$ for a real number $\xi$ greater than a certain constant. We prove that, from the asymptotic behavior, we can extract the $\rm{SL}(2;\mathbb{C})$ Chern--Simons invariant and the Reidemeister torsion twisted by the adjoint action both associated with a representation determined by $\xi$.