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

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 $\exp\xi$.