Asymptotics aspects of Teichm\"{u}ller 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 $\hbar \to 0^+$, 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.