The Andersen–Kashaev volume conjecture for the Teichmüller TQFT: existence and volume asymptotics of the Jones function $J_{M,K}$ for arbitrary hyperbolic knots in closed oriented 3-manifolds

For a hyperbolic knot $K$ in a closed oriented 3-manifold $M$, the Andersen–Kashaev volume conjecture predicts the existence of a smooth function $J_{M,K}(\hbar,x)$ on $\mathbb{R}_{>0}\times\mathbb{R}$ — the Jones function of the pair — that (1) reconstructs the Teichmüller TQFT partition function $Z_\hbar(X)$ of any fully balanced shaped ideal triangulation $X$ of $M\setminus K$ by a gauge-modified Laplace transform, (2) is recovered as a residue limit of the partition function on any one-vertex H-triangulation of $(M,K)$, and (3) satisfies $\lim_{\hbar\to 0}2\pi\hbar\log|J_{M,K}(\hbar,0)|=-\operatorname{vol}(M\setminus K)$, so the partition function decays exponentially at a rate encoding the hyperbolic volume. The conjecture is proved only for restricted classes of knots (figure-eight and $5_2$; all twist knots; $7_3$; all knots admitting FAMED geometric triangulations, including all knots up to 12 crossings; some lens-space families), and the general case — arbitrary hyperbolic knots in arbitrary closed oriented 3-manifolds, with the universal 'for any triangulation' clauses and the H-triangulation asymptotics — remains open. The problem is to prove, or refute, the conjecture in this generality.

Background

The Teichmüller TQFT of Andersen–Kashaev (2014) is a state-integral topological quantum field theory built from the quantum Teichmüller theory of a surface: its partition function $Z_\hbar(X)$ is defined as a multidimensional integral over edge variables of a product of Faddeev quantum dilogarithms $\hat\Phi_b$, one per tetrahedron, where $\hbar=(b+b^{-1})^{-2}$ and $b>0$ is the deformation parameter. The input is a shaped triangulation — an ideal triangulation (all vertices removed, e.g. the complement of a knot) in which each tetrahedron carries the dihedral angles of an ideal hyperbolic tetrahedron, with total angle $\pi$ per tetrahedron. An edge of a shaped triangulation is balanced if it is internal and the total dihedral angle around it is $2\pi$, and a shaped triangulation is fully balanced if all its edges are balanced; this balancing is what makes $Z_\hbar(X)$ independent (up to explicit gauge factors) of the chosen 2-3/3-2 Pachner moves relating such triangulations. An H-triangulation of a pair $(M,K)$ is a one-vertex triangulation of $M$ in which the knot appears as one edge, with dihedral angles $\omega_Y$ around the edges ($\tau$ records which edge is the knot: $\tau=0$ on the knot edge, $2\pi$ elsewhere). The Teichmüller TQFT is expected to be complex Chern–Simons theory for $\mathrm{PSL}(2,\mathbb{C})$ at level $k=1$, and its partition functions are explicitly computable, rigorously convergent integrals — unlike most physics TQFT partition functions. The Andersen–Kashaev volume conjecture posits that the semiclassical asymptotics of these partition functions universally encode hyperbolic volume: there should exist a triangulation-independent function $J_{M,K}(\hbar,x)$ (the Jones function, by analogy with the colored Jones polynomial, of which it is expected to be a nonperturbative completion) such that (1) for any fully balanced shaped ideal triangulation $X$ of $M\setminus K$, with $\lambda$ a gauge-invariant real linear combination of the dihedral angles and $\varphi$ a gauge non-invariant real quadratic polynomial in them, $Z_\hbar(X)=e^{i\varphi/\hbar}\int_{\mathbb{R}}J_{M,K}(\hbar,x)e^{-x\lambda/\sqrt{\hbar}}\,dx$; (2) for any one-vertex shaped H-triangulation $Y$, $\lim_{\omega_Y\to\tau}\hat\Phi_b\!\left(\frac{\pi-\omega_Y(K)}{2\pi i\sqrt{\hbar}}\right)Z_\hbar(Y)=e^{i\varphi/\hbar-i\pi/12}J_{M,K}(\hbar,0)$; (3) $\lim_{\hbar\to 0}2\pi\hbar\log|J_{M,K}(\hbar,0)|=-\operatorname{vol}(M\setminus K)$. The negative sign means decay rather than growth: unlike the Kashaev–Murakami–Murakami volume conjecture for the colored Jones polynomial, which predicts that the $N$-th colored Jones polynomial grows like $e^{\operatorname{vol}(M\setminus K)\,N/(2\pi)}$ as $N\to\infty$ and is proved only in special cases, the Teichmüller TQFT partition function itself decays like $e^{-\operatorname{vol}/(2\pi\hbar)}$, and the conjecture predicts a universal mechanism: volume is the leading semiclassical content of the state integral. A 'new formulation' via Weil–Gel'fand–Zak transforms on edge-face variables was announced by Andersen–Kashaev in 2013, claimed equivalent to the original and to allow unrestricted Pachner moves, but the proofs were postponed to a publication that has never appeared publicly (arXiv:1305.4291).

Problem Statement

Let $(M,K)$ be a pair consisting of a closed oriented 3-manifold $M$ and a hyperbolic knot $K\subset M$ (i.e. the complement $M\setminus K$ admits a complete hyperbolic metric of finite volume). Prove or refute: there exists a smooth function $J_{M,K}:\mathbb{R}_{>0}\times\mathbb{R}\to\mathbb{C}$ such that, with $\hbar=(b+b^{-1})^{-2}$: (1) for any fully balanced shaped ideal triangulation $X$ of $M\setminus K$, there exist a gauge-invariant real linear combination $\lambda\in\mathbb{R}$ of the dihedral angles and a real quadratic polynomial $\varphi\in\mathbb{R}$ of the dihedral angles such that the Teichmüller TQFT partition function admits the factorization $Z_\hbar(X)=e^{i\varphi/\hbar}\int_{\mathbb{R}}J_{M,K}(\hbar,x)e^{-x\lambda/\sqrt{\hbar}}\,dx$ (the same $J_{M,K}$ must serve every such triangulation, each with its own $\lambda$ and $\varphi$); (2) for any one-vertex shaped H-triangulation $Y$ of the pair $(M,K)$, there exists a real quadratic polynomial $\varphi\in\mathbb{R}$ of the dihedral angles such that $\lim_{\omega_Y\to\tau}\hat\Phi_b\!\left(\frac{\pi-\omega_Y(K)}{2\pi i\sqrt{\hbar}}\right)Z_\hbar(Y)=e^{i\varphi/\hbar-i\pi/12}J_{M,K}(\hbar,0)$, where $\omega_Y$ assigns to each edge of $Y$ the sum of the dihedral angles around that edge, $\omega_Y(K)$ is the total dihedral angle around the knot edge, $\hat\Phi_b$ is Faddeev's quantum dilogarithm, and $\tau$ is the angle assignment taking the value $0$ on the knot edge and $2\pi$ on all other edges; and (3) the volume asymptotic $\lim_{\hbar\to 0}2\pi\hbar\log|J_{M,K}(\hbar,0)|=-\operatorname{vol}(M\setminus K)$ holds. The fully general statement, including the universal quantifiers over all fully balanced shaped ideal triangulations and all one-vertex shaped H-triangulations, and ambient manifolds beyond $S^3$ and lens spaces (in particular complements with $H_2\neq 0$), is the open target.

References

1. Jørgen Ellegaard Andersen, Rinat Kashaev, 'A TQFT from Quantum Teichmüller Theory', Communications in Mathematical Physics 330(3):887–934 (2014), doi:10.1007/s00220-014-2073-2, https://arxiv.org/abs/1109.6295
2. Jørgen Ellegaard Andersen, Jens-Jakob Kratmann Nissen, 'Asymptotic aspects of the Teichmüller TQFT', Travaux mathématiques 25:41–95, Université du Luxembourg (2017), arXiv:1612.06982, https://arxiv.org/abs/1612.06982
3. Jørgen Ellegaard Andersen, Rinat Kashaev, 'A new formulation of the Teichmüller TQFT' (2013), arXiv:1305.4291, https://arxiv.org/abs/1305.4291
4. Fathi Ben Aribi, François Guéritaud, Eiichi Piguet-Nakazawa, 'Geometric triangulations and the Teichmüller TQFT volume conjecture for twist knots', Quantum Topology 14(2):285–406 (2023), doi:10.4171/qt/178, https://arxiv.org/abs/1903.09480
5. Soichiro Uemura, 'A proof of the Teichmüller TQFT volume conjecture for $7_3$ knot' (2023), arXiv:2307.12848, https://arxiv.org/abs/2307.12848
6. Fathi Ben Aribi, Ka Ho Wong, 'The Andersen-Kashaev volume conjecture for FAMED geometric triangulations' (2024), arXiv:2410.10776, https://arxiv.org/abs/2410.10776
7. Fathi Ben Aribi, Antonin Guilloux, Ka Ho Wong, 'FAMED by computer: proving the Andersen-Kashaev volume conjecture for 42,000 knots' (2025), arXiv:2512.17437, https://arxiv.org/abs/2512.17437
8. Ka Ho Wong, 'Asymptotics aspects of Teichmüller TQFT for generalized FAMED semi-geometric triangulations' (2025), arXiv:2512.23198, https://arxiv.org/abs/2512.23198
9. Jørgen Ellegaard Andersen, Rinat Kashaev, 'The Teichmüller TQFT', Proceedings of the International Congress of Mathematicians 2018, World Scientific:2541–2565 (2019), doi:10.1142/9789813272880_0149, https://arxiv.org/abs/1811.06853
10. Jørgen Ellegaard Andersen, Alessandro Malusà, 'A geometric quantisation view on the AJ-conjecture for the Teichmüller TQFT', Research in the Mathematical Sciences 13(1), article 4 (2025), doi:10.1007/s40687-025-00585-9, https://arxiv.org/abs/1711.11522