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

We study the asymptotic behavior of the $N$-dimensional colored Jones polynomial of the figure-eight knot evaluated at $\exp ((\kappa +2p\pi \mathrm{ı}\mathrm{/}N)$, where $\kappa :=\text{arccosh}(3\mathrm{/}2)$ 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 $\text{SL}(2;\text{C})$.