The Andersen-Kashaev volume conjecture for FAMED geometric triangulations

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 Teichmüller 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 Teichmüller TQFT developed by Andersen-Malusa. We further discover a new phenomenon: for FAMED geometric triangulations, the partition function in Teichmüller 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 Guéritaud, 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 $S^3$.

💡 Research Summary

The paper introduces a new combinatorial class of ideal triangulations called FAMED (Face‑Adjacency‑Matrices‑with‑Edge‑Duality) and uses it to settle the Andersen‑Kashaev volume conjecture for a broad family of one‑cusped hyperbolic 3‑manifolds.

A FAMED triangulation X of a manifold M (with one torus boundary and trivial second homology) is defined by four conditions: (i) the space of angle structures A_X is non‑empty; (ii) the face‑adjacency matrix A has zero determinant; (iii) the Neumann‑Zagier edge matrix B also has zero determinant; and (iv) the matrices satisfy the duality relation
 B⁻¹A = X₀A⁻¹B + (X₀A⁻¹B)ᵀ + E + I_N,
where X₀ encodes the edge‑to‑vertex incidence, E records the signs of tetrahedra, and I_N is the identity. This relation ties together the combinatorics of faces with the algebra of edge gluing equations and is the heart of the subsequent analysis.

With a FAMED triangulation in hand, the authors construct the Teichmüller TQFT partition function Z_ℏ(X,α) for any angle structure α∈A_X. The construction uses Faddeev’s quantum dilogarithm and the matrices A, B, E, yielding an explicit integral expression. Applying a rigorous saddle‑point (steepest‑descent) analysis as ℏ→0⁺, they obtain an asymptotic expansion \