On the asymptotic expansions of various quantum invariants I: the colored Jones polynomial of twist knots at the root of unity $e^{\frac{2\pi\sqrt{-1}}{N+\frac{1}{2}}}$

TL;DR

This paper derives an asymptotic expansion formula for the colored Jones polynomial of twist knots at specific roots of unity, enhancing understanding of quantum invariants' behavior in the large color limit.

Contribution

It introduces a new asymptotic expansion formula for the colored Jones polynomial of twist knots at roots of unity, using the saddle point method.

Findings

Derived explicit asymptotic expansion formula

Applied saddle point method to quantum invariants

Enhanced understanding of quantum invariants at roots of unity

Abstract

This is the first 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, we obtain an asymptotic expansion formula for the colored Jones polynomial of twist knots $\mathcal{K}_p$ with $p\geq 6$ at the root of unity $e^{\frac{2\pi\sqrt{-1}}{N+\frac{1}{2}}}$.