Asymptotics aspects of Teichm\"{u}ller TQFT for generalized FAMED semi-geometric triangulations

TL;DR

This paper studies the asymptotic behavior of Teichmüller TQFT partition functions for certain ideal triangulations of hyperbolic knot complements, linking decay rates to hyperbolic volume and invariants like the Jones polynomial.

Contribution

It introduces a generalized FAMED property for triangulations, proves exponential decay of the partition function in the semi-classical limit, and confirms the Andersen-Kashaev volume conjecture for specific triangulations.

Findings

Partition function decays exponentially with hyperbolic volume.

1-loop invariant appears as the 1-loop term in the asymptotics.

Under certain conditions, the Jones function's decay rate is governed by the Neumann-Zagier potential.

Abstract

We introduce a generalized FAMED property for ideal triangulations of hyperbolic knot complements in $\mathbb{S}^3$. Given a hyperbolic knot $K$ in $\mathbb{S}^3$ and a semi-geometric triangulation $X$ of $\mathbb{S}^3 \setminus K$ that is generalized FAMED with respect to the longitude. We prove that in the semi-classical limit $\hbar \to 0^+$, for any angle structure $\alpha$, the partition function $\mathscr{Z}_\hbar(X,\alpha)$ in Teichm\"uller TQFT decays exponentially with decrease rate the volume of $\mathbb{S}^3 \setminus K$ equipped with a hyperbolic cone structure determined by $\alpha$, and that the 1-loop invariant of Dimofte-Garoufalidis emerges as the 1-loop term. With additional combinatorial conditions on the triangulations, we prove the existence of the Jones function and show that its decay rate is governed by the Neumann-Zagier potential function. In particular, the Andersen-Kashaev volume conjecture holds for every hyperbolic knot whose complement admits such kinds of triangulations.