Algebra and Combinatorics Seminar

Date: February 23, 2018

Time: 3:00PM - 4:00PM

Location: BLOC 628

Speaker: Ka Ho Wong, Chinese University of Hong Kong

Title: Asymptotic expansion formula for the colored Jones polynomial and Turaev-Viro invariant for the figure eight knot

Abstract: The volume conjecture of the Turaev-Viro invariant is a new topic in quantum topology. It has been shown that the $(2N+1)$-th Turaev-Viro invariant for the knot complement $\SS^3 \backslash K$ can be expressed as a sum of the colored Jones polynomial of $K$ evaluated at $\exp(2\pi i/ (N+1/2))$. That leads to the study of the asymptotic expansion formula (AEF) for the colored Jones polynomial of $K$ evaluated at half-integer root of unity. When $K$ is the figure eight knot, by using saddle point approximation, H.Murakami had already found out the AEF for the $N$-th colored Jones polynomial of $K$ evaluated at $\exp(2\pi i/N)$. In this talk, I will first review the strategy Murakami used to prove the AEF of the colored Jones polynomial. Then, I will further discuss, for $M$ with a fixed limiting ratio of $M$ and $(N+1/2)$, how the AEF for the $M$-th colored Jones polynomial for the figure eight knot evaluated at $(N+1/2)$-th root of unity can be obtained. As an application of the asymptotic behavior of the colored Jones polynomials mentioned above, we obtain the asymptotic expansion formula for the Turaev-Viro invariant 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.