The Andersen-Kashaev volume conjecture for FAMED geometric triangulations

Abstract

We investigate the Andersen-Kashaev volume conjecture by introducing the notion of FAMED triangulations, a class of ideal triangulations of $3$-manifolds satisfying certain specific combinatorial properties. For any FAMED triangulation of a one-cusped hyperbolic $3$-manifold $M$ with trivial second homology, we prove the existence of the Jones function in the Teichm\"uller TQFT of $M$. For FAMED geometric triangulations of $M$, we establish an asymptotic expansion of the Jones function in terms of the Neumann-Zagier potential function and the 1-loop invariant of Dimofte-Garoufalidis. As a consequence, we prove the Andersen-Kashaev volume conjecture for $M$ and provide new insights for the AJ conjecture for the Teichm\"uller TQFT developed by Andersen-Malusa. We further discover a new phenomenon: for FAMED geometric triangulations, the partition function in Teichm\"uller TQFT decays exponentially with decrease rate the hyperbolic volume of a cone structure determined by the prescribed angle structure. This perspective provides a potential application to the Casson conjecture on angle structures. Expanding the previous result of Gu\'eritaud, Piguet-Nakazawa and the first author and complementing a parallel result of Guilloux and both authors, we prove all the above generalizations of the Andersen-Kashaev volume conjecture for every hyperbolic twist knot and for the first 42,000 hyperbolic knots in $S^3$.