Asymptotics aspects of Teichmüller TQFT for generalized FAMED semi-geometric triangulations

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 $\hslash \to 0^{+}$, for any angle structure $\alpha$, the partition function ${\mathcal{Z}}_{\hslash }(X,\alpha )$ in Teichmüller 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.

47 pages, 4 figures. arXiv admin note: text overlap with arXiv:2410.10776