The asymptotic behaviors of the colored Jones polynomials of the figure eight-knot, and an affine representation

TL;DR

This paper investigates the asymptotic growth of the colored Jones polynomial for the figure-eight knot at specific complex values, linking it to the Chern--Simons invariant of an affine representation.

Contribution

It establishes a connection between the polynomial's exponential growth rate and the Chern--Simons invariant of an affine SL(2,C) representation.

Findings

Exponential growth of the polynomial with specific rate

Growth rate determined by the Chern--Simons invariant

Links quantum invariants to geometric structures

Abstract

We study the asymptotic behavior of the $N$-dimensional colored Jones polynomial of the figure-eight knot evaluated at $\exp\bigl((\kappa+2p\pi\i/N\bigr)$, where $\kappa:=\arccosh(3/2)$ and $p$ is a positive integer. We can prove that it grows exponentially with growth rate determined by the Chern--Simons invariant of an affine representation from the fundamental group of the knot complement to the Lie group $\SL(2;\C)$.