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The Andersen-Kashaev volume conjecture for FAMED geometric triangulations

Fathi Ben Aribi, Ka Ho Wong

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 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 $S^3$.

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 -manifolds satisfying certain specific combinatorial properties. For any FAMED triangulation of a one-cusped hyperbolic -manifold with trivial second homology, we prove the existence of the Jones function in the Teichmüller TQFT of . For FAMED geometric triangulations of , 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 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 .

Paper Structure

This paper contains 33 sections, 35 theorems, 241 equations, 4 figures.

Table of Contents

  1. Introduction
  2. FAMED ideal triangulation and volume conjectures
  3. Asymptotics of Jones functions
  4. Jones functions and the AJ-conjecture
  5. Asymptotics of the Teichmüller TQFT partition functions
  6. Approaching the Casson conjecture via FAMED triangulations
  7. Computational results about FAMED triangulations
  8. Outline of the paper
  9. Preliminaries
  10. Triangulations
  11. Angle structures
  12. Thurston's complex gluing equations
  13. The classical dilogarithm
  14. The Bloch--Wigner function
  15. Faddeev's quantum dilogarithm
  16. ...and 18 more sections

Key Result

Theorem 1.6

Let $M$ be a one-cusped hyperbolic $3$-manifold with trivial second homology and let $X$ be an ordered ideal triangulation of $M$.

Figures (4)

  • Figure 1: Thurston's triangulation of $M=S^3 {\smallsetminus} 4_1$, and the face adjacency matrices
  • Figure 2: Cusp triangulation of $S^3 {\smallsetminus} 4_1$, and the Neumann-Zagier matrices
  • Figure 3: The positive tetrahedron $T$
  • Figure 4: On the left hand side, we have an ideal tetrahedron with shape parameters $z_i, z_i'$ and $z_i"$ assigned to the edges. In the middle, the black dot with a label $e_i$ corresponds to the $i$-th edge, which is surrounded by truncated triangles around the ideal vertices of the tetrahedra in the triangulation. In this example, $E_{ij} = 2$. On the right hand side, the rectangle is a fundamental domain of the boundary torus $\mathbb{T}_i$ and $\alpha_i$ is a lifting of the simple closed curve $\alpha_i \in \pi_1(\mathbb{T}_i)$ to the fundamental domain. In this example, we have $C_{ij} = 1 - 1 = 0$.

Theorems & Definitions (85)

  • Definition 1.1
  • Remark 1.2
  • Remark 1.3
  • Conjecture 1.5: see AK, Conjecture 1.(1)&(3) and BAGPN Conjecture 2.13
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Corollary 1.9
  • Theorem 1.10
  • Corollary 1.11
  • ...and 75 more