Asymptotic aspects of the Teichm\"uller TQFT

TL;DR

This paper computes knot invariants from Teichmüller TQFT, verifies their equivalence across formulations, and connects the invariants to the Andersen-Kashaev volume conjecture through stationary phase analysis.

Contribution

It demonstrates the equivalence of two formulations of Teichmüller TQFT for knot invariants and links these invariants to the volume conjecture, providing explicit isomorphisms.

Findings

Equivalent results from two Teichmüller TQFT formulations.

Verification of the Andersen-Kashaev volume conjecture.

Explicit isomorphism between TQFT representation and Schwartz functions.

Abstract

We calculate the knot invariant coming from the Teichm\"{u}ller TQFT [AK1]. Specifically we calculate the knot invariant for the complement of the knot $6_1$ both in the original [AK1] and the new formulation of the Teichm\"{u}ller TQFT [AK2] for the one-vertex H-triangulation of $(S^3,6_1)$. We show that the two formulations give equivalent answers. Furthermore we apply a formal stationary phase analysis and arrive at the Andersen- Kashaev volume conjecture as stated in [AK1, Conj. 1]. Furthermore we calculate the first examples of knot complements in the new formulation showing that the new formulation is equivalent to the original one in all the special cases considered. Finally, we provide an explicit isomorphism between the Teichm\"{u}ller TQFT representation of the mapping class group of a once punctured torus and a representation of this mapping class group on the space of Schwartz class functions on the real line.