The Andersen-Kashaev volume conjecture for FAMED geometric triangulations

TL;DR

This paper proves the Andersen-Kashaev volume conjecture for a broad class of hyperbolic 3-manifolds using FAMED triangulations, linking quantum invariants to hyperbolic volume and exploring new phenomena in Teichmüller TQFT.

Contribution

Introduction of FAMED triangulations and proof of the Andersen-Kashaev volume conjecture for various hyperbolic knots and manifolds, expanding the conjecture's scope.

Findings

Proved the conjecture for all hyperbolic twist knots.

Extended the conjecture to the first 42,000 hyperbolic knots in S^3.

Discovered exponential decay of the partition function related to hyperbolic volume.

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 $S^3$.