The colored Jones polynomial of the figure-eight knot and an $\mathrm{SL}(2;\mathbb{R})$-representation

Abstract

We study the asymptotic behavior of the $N$-dimensional colored Jones polynomial of the figure-eight knot, evaluated at $\exp ((u+2p\pi \sqrt{-1})/N)$ as $N$ tends to infinity, where $u>\mathrm{arccosh}(3/2)$ is a real number and $p\ge 1$ is an integer. It turns out that it corresponds to an $\mathrm{SL}(2;\mathbb{R})$ representation of the fundamental group of the knot complement. Moreover, it defines the adjoint Reidemeister torsion and the Chern--Simons invariant associated with the representation.

45 pages, 6 figures