The generalized volume conjecture for the figure-eight knot parametrized by a complex number with small imaginary part

TL;DR

This paper investigates the asymptotic behavior of the colored Jones polynomial of the figure-eight knot evaluated at complex parameters with small imaginary parts, revealing connections to the Chern--Simons invariant and Alexander polynomial.

Contribution

It extends the volume conjecture to a complex parameter regime, establishing new asymptotic relations involving the Chern--Simons invariant and Alexander polynomial.

Findings

Exponential growth rate linked to the Chern--Simons invariant for large real part of .

Convergence to reciprocal of the Alexander polynomial for small real part of .

Provides a unified view of the asymptotic behavior across different parameter regimes.

Abstract

We study the asymptotic behavior, as $N$ tends to infinity, of the $N$-dimensional colored Jones polynomial of the figure-eight knot, evaluated at $\exp(\xi/N)$ for a complex parameter $\xi$ with $0<\mathrm{Im}\xi<\pi/2$. We prove that if $\mathrm{Re}{\xi}$ is large the colored Jones polynomial grows exponentially with growth rate expressed by the Chern--Simons invariant, and that if $\mathrm{Re}{\xi}$ is small it converges to the reciprocal of the Alexander polynomial evaluated at $\exp\xi$.