Volume 22, issue 1 (2022)

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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

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+1 2) th roots of unity with a fixed limiting ratio, s, of M and N + 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 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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volume conjecture, figure eight knot, colored Jones polynomials, Turaev–Viro invariants
Mathematical Subject Classification 2010
Primary: 57M27
Secondary: 57M50
Received: 14 June 2018
Revised: 21 September 2018
Accepted: 30 October 2018
Published: 26 April 2022
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
Hong Kong