On the asymptotic expansion of quantum invariants related to surgeries of Whitehead link I: Relative Reshetikhin-Turaev invariants and the Turaev-Viro invariants at $e^{\frac{2\pi\sqrt{-1}}{N+\frac{1}{2}}}$

TL;DR

This paper derives asymptotic expansion formulas for relative Reshetikhin-Turaev and Turaev-Viro invariants of 3-manifolds obtained via rational surgery on Whitehead link components, advancing understanding of quantum invariants in topology.

Contribution

It provides the first asymptotic expansion formulas for these invariants in the context of Whitehead link surgeries, linking quantum invariants to geometric structures.

Findings

Asymptotic expansion formulas for relative Reshetikhin-Turaev invariants.

Asymptotic expansion formulas for Turaev-Viro invariants.

Application to 3-manifolds obtained by Whitehead link surgeries.

Abstract

In this article, we obtain an asymptotic expansion formula for the relative Reshetikhin-Turaev invariant in the case that the ambient 3-manifold is gained by doing rational surgery along one component of Whitehead link. In addition, we obtain an asymptotic expansion formula for the Turaev-Viro invariant of the cusped 3-manifold which is gained by doing rational surgery along one component of the Whitehead link.