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

Abstract

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

58 pages, 21 figures

References in corpus (4)

• Three-Dimensional Quantum Gravity, Chern-Simons Theory, and the A-Polynomial
• Skein-theoretical derivation of some formulas of Habiro
• The colored Jones polynomial, the Chern--Simons invariant, and the Reidemeister torsion of the figure-eight knot
• The colored Jones polynomial of the figure-eight knot and a quantum modularity