The colored Jones polynomial of the figure-eight knot and a quantum modularity

Abstract

We study the asymptotic behavior of the $N$-dimensional colored Jones polynomial of the figure-eight knot evaluated at $\exp ((u+2p\pi ı)/N)$, where $u$ is a small real number and $p$ is a positive integer. We show that it is asymptotically equivalent to the product of the $p$-dimensional colored Jones polynomial evaluated at $\exp (4N{\pi }^2/(u+2p\pi ı))$ and a term that grows exponentially with growth rate determined by the Chern--Simons invariant. This indicates a quantum modularity of the colored Jones polynomial.

31 pages, 7 figures

References in corpus (5)

• Skein-theoretical derivation of some formulas of Habiro
• Asymptotics of the colored Jones function of a knot
• The colored Jones polynomial, the Chern--Simons invariant, and the Reidemeister torsion of the figure-eight knot
• Proof of the volume conjecture for Whitehead doubles of a family of torus knots
• The colored Jones polynomials and the Alexander polynomial of the figure-eight knot

Cited by in corpus (1)

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