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

TL;DR

This paper derives asymptotic expansion formulas for the colored Jones polynomial of twist knots at specific roots of unity, extending previous results and analyzing the limit behavior as parameters vary.

Contribution

It provides new asymptotic expansion formulas for the colored Jones polynomial of twist knots at roots of unity, including a limit case, advancing understanding of quantum invariants.

Findings

Asymptotic expansion formula at $e^{2\pi i/(N+1/M)}$ for $p extgreater=6$

Limit formula as $M o+\infty$ at $e^{2\pi i/N}$

Extension of previous methods to new root of unity cases

Abstract

This is the second article in a series devoted to the study of the asymptotic expansions of various quantum invariants related to the twist knots. In this article, following the method and results in \cite{CZ23-1}, we present an asymptotic expansion formula for the colored Jones polynomial of twist knot $\mathcal{K}_p$ with $p\geq 6$ at the root of unity $e^{\frac{2\pi\sqrt{-1}}{N+\frac{1}{M}}}$ with $M\geq 2$. Furthermore, by taking the limit $M\rightarrow +\infty$, 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 $e^{\frac{2\pi\sqrt{-1}}{N}}$.