On the asymptotic behavior of the colored Jones polynomial of the figure-eight knot associated with a real number

Abstract

We study the asymptotic behavior of the $N$-dimensional colored Jones polynomial evaluated at $\exp (\xi /N)$ for a real number $\xi$ greater than a certain constant. We prove that, from the asymptotic behavior, we can extract the $\mathrm{SL}(2;\mathbb{C})$ Chern--Simons invariant and the Reidemeister torsion twisted by the adjoint action both associated with a representation determined by $\xi$.

arXiv comment27 pages, 1 figure: typos corrected, Remark 1.6 added, Lemma 6.3 added

References in corpus (6)

• Discrete Heisenberg-Weyl Group and Modular Group
• Skein-theoretical derivation of some formulas of Habiro
• Asymptotics of the colored Jones function of a knot
• An introduction to the volume conjecture and its generalizations
• The colored Jones polynomial, the Chern--Simons invariant, and the Reidemeister torsion of the figure-eight knot
• Some limits of the colored Jones polynomials of the figure-eight knot