Asymptotic behavior of the colored Jones polynomials and Turaev–Viro invariants of the figure eight knot

Wong, Ka Ho; Au, Thomas Kwok-Keung

doi:10.2140/agt.2022.22.1

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Asymptotic behavior of the colored Jones polynomials and Turaev–Viro invariants of the figure eight knot

Ka Ho Wong and Thomas Kwok-Keung Au

Algebraic & Geometric Topology 22 (2022) 1–53

DOI: 10.2140/agt.2022.22.1

Abstract

We investigate the asymptotic behavior of the colored Jones polynomials and the Turaev–Viro invariants for the figure eight knot. More precisely, we consider the $M^{th}$ colored Jones polynomials evaluated at ${(N + \frac{1}{2})}^{th}$ roots of unity with a fixed limiting ratio, $s$ , of $M$ and $N + \frac{1}{2}$ . We find out the asymptotic expansion formula (AEF) of the colored Jones polynomials of the figure eight knot with $s$ close to $1$ . We show that the exponential growth rate of the colored Jones polynomials of the figure eight knot with $s$ close to $\frac{1}{2}$ is strictly less than those with $s$ close to $1$ . It is known that the Turaev–Viro invariant of the figure eight knot can be expressed in terms of a sum of its colored Jones polynomials. Our results show that this sum is asymptotically equal to the sum of the terms with $s$ close to $1$ . As an application of the asymptotic behavior of the colored Jones polynomials, we obtain the asymptotic expansion formula for the Turaev–Viro invariants of the figure eight knot. Finally, we suggest a possible generalization of our approach so as to relate the AEF for the colored Jones polynomials and the AEF for the Turaev–Viro invariants for general hyperbolic knots.

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Keywords

volume conjecture, figure eight knot, colored Jones polynomials, Turaev–Viro invariants

Mathematical Subject Classification 2010

Primary: 57M27

Secondary: 57M50

References

Publication

Received: 14 June 2018

Revised: 21 September 2018

Accepted: 30 October 2018

Published: 26 April 2022

Authors

Ka Ho Wong
Department of Mathematics Texas A&M University College Station, TX United States

Thomas Kwok-Keung Au
Department of Mathematics The Chinese University of Hong Kong Shatin Hong Kong

Volume 22, issue 1 (2022)