On the asymptotic expansion of various quantum invariants III: the Reshetikhin-Turaev invariants of closed hyperbolic 3-manifolds obtained by doing integral surgery along the twist knot

TL;DR

This paper derives an asymptotic expansion formula for Reshetikhin-Turaev invariants of certain hyperbolic 3-manifolds created via integral surgery on twist knots, using saddle point methods.

Contribution

It provides a new asymptotic expansion formula for quantum invariants of hyperbolic 3-manifolds obtained through integral surgery on twist knots.

Findings

Asymptotic expansion formula for Reshetikhin-Turaev invariants

Application of saddle point method to quantum invariants

Results for invariants at roots of unity

Abstract

This is the third article in a series devoted to the study of the asymptotic expansions of various quantum invariants related to the twist knots. In this paper, by using the saddle point method developed by Ohtsuki and Yokota, we obtain an asymptotic expansion formula for the Reshetikhin-Turaev invariants of closed hyperbolic 3-manifolds obtained by doing integral $q$-surgery along the twist knots $\mathcal{K}_p$ at the root of unity $e^{\frac{4\pi\sqrt{-1}}{r}}$ ($r$ is odd).