Asymptotics aspects of Teichmüller TQFT for generalized FAMED semi-geometric triangulations

TL;DR

This work generalizes the asymptotic analysis of Teichmüller TQFT invariants to generalized FAMED semi-geometric triangulations of hyperbolic knot complements. It develops a robust saddle-point framework built on Neumann– Zagier data and a generalized FAMED condition, deriving precise asymptotics for partition functions and linking them to hyperbolic volumes and 1-loop torsion. Under additional combinatorial hypotheses, the paper establishes the existence and asymptotics of Jones functions, thereby proving the Andersen–Kashaev volume conjecture for knots admitting such triangulations. The study blends analytic continuation of dilogarithms, complex Morse theory, and contour deformation to connect quantum invariants with classical hyperbolic geometry through the Neumann– Zagier potential and 1-loop data. Together, these results provide a unified, geometrically informed pathway from Teichmüller TQFT partitions to volume conjectures in hyperbolic 3-manifolds.

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 $α$, the partition function $\mathscr{Z}_\hbar(X,α)$ in Teichmüller TQFT decays exponentially with decrease rate the volume of $\mathbb{S}^3 \setminus K$ equipped with a hyperbolic cone structure determined by $α$, 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.

Figures (4)

• Figure 1: Thurston's triangulation of $M={\mathbb S}^3 {\smallsetminus} 4_1$, and the face adjacency matrices
• Figure 2: The first and second figures respectively show the combinatorics of the triangulation that is allowed and disallowed in Definition \ref{['defgenFAMED5']}(2). In the first figure, the edge imposes a linear equation $\operatorname{Log} z_1 + \operatorname{Log} z_2 + \dots + \operatorname{Log} z_5 = 2\pi i$ to the variables $\operatorname{Log} z_1,\dots, \operatorname{Log} z_N$. Note that the equation is independent of the choice of the logarithmic holonomy of the longitude $\xi$. In contrast, in the second figure, if the red curve represents the longitude of the knot on the cusp triangulation of the boundary torus, then the holonomy equation is given by $\operatorname{Log} z_1 + \dots + \operatorname{Log} z_6 = \xi$, which depends on the choice of $\xi$.
• Figure 3: This figure shows a schematic picture of the contours constructed in Proposition \ref{['constructZcont']}, \ref{['Zcont']} and used in the proof of Theorem \ref{['mainthmZ']}. In Proposition \ref{['constructZcont']}, we first deform the horizontal contour $\tilde{L}^{\text{top}}$ by pushing it upward around the singularity $-i\pi$ and the boundary points. Then we further deform the contour by following the flow generated by $\mathscr{V}$ that decreases the value of $\operatorname{Re} S(\boldsymbol{x}; \lambda_X(\alpha))$. The critical point is stationary in this process and the resulting red contour is $L^\text{top}$. The blue contour represents $({\mathbb R}^{N-2n}+ i \mathbf{v}_\alpha) {\smallsetminus} ( (-\kappa,\kappa)^{N-2n} + i \mathbf{v}_\alpha)$. We connect the red and blue contours through $L^{\text{sides}}$, which is colored in green in the figure. Note that the union of these three contours is homotopic to $({\mathbb R}^{N-2n}+ i \mathbf{v}_\alpha)$, which is the integration contour of the partition function.
• Figure 4: This figure shows a schematic picture of the contours constructed in Proposition \ref{['Jcont2']} and used in the proof of Theorem \ref{['thm:Jones:genFAMED']}. As shown in the right figure, Complex Morse Lemma provides a nice local coordinate chart for us to deform the black contour to the maroon contour that passes through the critical point (the origin).

Theorems & Definitions (94)

• Definition 1.1
• Conjecture 1.2
• Remark 1.3
• Remark 1.4
• Definition 1.5
• Remark 1.6
• Remark 1.7
• Conjecture 1.8
• Remark 1.9
• Remark 1.10