The Andersen–Kashaev volume conjecture for the Teichmüller TQFT (general hyperbolic knots in closed 3-manifolds)

The Teichmüller topological quantum field theory (TQFT) constructed by Andersen and Kashaev from quantum Teichmüller theory assigns to each admissible shaped ideal triangulation of a 3-manifold a partition function built from Faddeev's quantum dilogarithm, and is expected to realize $SL(2,\mathbb C)$ Chern–Simons theory. The Andersen–Kashaev volume conjecture predicts that for every hyperbolic knot $K$ in a closed oriented 3-manifold $M$ there is a canonical 'Jones function' $J_{M,K}(\hbar,x)$, an analogue of the Kashaev invariant, such that the partition functions of ideal triangulations of $M\setminus K$ are Laplace-type transforms of $J_{M,K}$, partition functions of one-vertex H-triangulations of $(M,K)$ converge to $J_{M,K}(\hbar,0)$ up to an explicit phase, and $2\pi\hbar\log|J_{M,K}(\hbar,0)|\to-\mathrm{Vol}(M\setminus K)$ as $\hbar\to 0^+$. The conjecture has been proved only for restricted classes: the knots $4_1$, $5_2$, $6_1$, all twist knots, the knot $7_3$, infinite families of knots in lens spaces, and — via the FAMED (Face Adjacency Matrices with Edge Duality) framework of Ben Aribi and Wong — all one-cusped hyperbolic 3-manifolds with trivial second homology admitting a FAMED geometric triangulation, including 42,117 knots of the SnapPy knot census (all hyperbolic knots up to 12 crossings). The general conjecture, in particular for arbitrary hyperbolic knots in $S^3$ and for knots in closed manifolds other than $S^3$, remains open.

Background

The original volume conjecture (Kashaev 1997; Murakami–Murakami 2001) predicts that the hyperbolic volume of the complement of a hyperbolic knot $K\subset S^3$ — the volume of its unique complete hyperbolic metric — is encoded in the exponential decay of the colored Jones polynomial (the Jones polynomial of the $N$-dimensional irreducible representation of $SL(2,\mathbb C)$, evaluated at the primitive $N$-th root of unity) as $N\to\infty$. It has been proved rigorously only for several knots and special families, and remains one of the central open problems connecting quantum topology and hyperbolic geometry.

Andersen and Kashaev constructed, via quantum Teichmüller theory (the quantization of Teichmüller space of punctured surfaces), a generalized TQFT — the Teichmüller TQFT — that is expected to correspond to $SL(2,\mathbb C)$ Chern–Simons theory. Its input is combinatorial: a shaped ideal triangulation, i.e. a decomposition of a 3-manifold into ideal tetrahedra (tetrahedra with their vertices removed) glued along faces, where each tetrahedron carries shape parameters, equivalently dihedral angles, satisfying Thurston's gluing equations. The shape structure defines a weight function $\omega_X(e)$, the sum of the dihedral angles around each edge $e$, and a shaped triangulation is called fully balanced when $\omega_X(e)=2\pi$ for every edge. Admissibility requires positive dihedral angles and vanishing of $H_2(X-\Delta_0(X),\mathbb Z)$, where $\Delta_0(X)$ denotes the vertex set of $X$. The partition function $Z_\hbar(X)$ is a state integral: a contour integral over one real variable per tetrahedron whose integrand is a product of copies of Faddeev's quantum dilogarithm $\Phi_b(z)$, a meromorphic special function satisfying a functional equation that deforms the classical five-term dilogarithm identity, with the semiclassical parameter $\hbar=(b+b^{-1})^{-2}$. For pairs $(M,K)$ one also uses one-vertex H-triangulations: triangulations of $M$ with a single vertex in which the knot $K$ appears as an edge of a single tetrahedron.

Andersen and Kashaev proposed a volume conjecture for this TQFT (Conjecture 2.22 in Uemura's numbering, from Andersen–Kashaev 2014): for every hyperbolic knot $K$ in a closed oriented 3-manifold $M$ there exists a smooth 'Jones function' $J_{M,K}(\hbar,x)$ on $\mathbb R_{>0}\times\mathbb R$, an analogue of the Kashaev invariant of $K$, such that: (1) for any fully balanced shaped ideal triangulation $X$ of the complement $M\setminus K$, there are a real linear combination $\lambda$ of gauge-invariant dihedral angles (functionals invariant under the natural gauge-equivalence action on the space of shape structures) and a real second-order polynomial $\phi$ of dihedral angles such that $Z_\hbar(X)$ equals $e^{i\phi/\hbar}$ times a Laplace-type integral of $J_{M,K}$ against the weight $e^{-x\lambda/\sqrt{\hbar}}$; (2) for any one-vertex shaped H-triangulation $Y$ of $(M,K)$, multiplying the partition function by the quantum dilogarithm $\Phi_b$ evaluated at the angular shift $(\pi-\omega_Y(K))/(2\pi i\sqrt{\hbar})$, where $\omega_Y(K)$ is the weight of the edge representing $K$, and letting the shape structure degenerate to the one whose weight function takes the value $0$ on the edge representing $K$ and $2\pi$ on all other edges, the partition function converges to $e^{i\varphi/\hbar-i\pi/12}J_{M,K}(\hbar,0)$; and (3) $\lim_{\hbar\to 0^+}2\pi\hbar\log|J_{M,K}(\hbar,0)|=-\mathrm{Vol}(M\setminus K)$, so the hyperbolic volume of the knot complement is exactly the exponential decay rate of the Jones function.

Prior to the source paper, the conjecture was known only for $4_1$ and $5_2$ (Andersen–Kashaev 2014, for specific triangulations), for $6_1$ (Andersen–Nissen 2016), for all twist knots (Ben Aribi–Guéritaud–Piguet-Nakazawa, Quantum Topology 2023) — the latter in a reformulated form, for explicit geometric triangulations and $b>0$, obtained by rigorous saddle-point evaluation of the state integrals — and for infinite families of knots in lens spaces (Piguet-Nakazawa's thesis, extending the twist-knot results to closed manifolds other than $S^3$). Uemura (2023) added the knot $7_3$, which is not a twist knot, in the same reformulated sense: properties (1)–(2) were confirmed for one specific ideal triangulation and one specific one-vertex H-triangulation of $(S^3,7_3)$ (with a contour intentionally different from the conjecture's), and the volume asymptotics (3) was proved by stationary-phase analysis with rigorous error bounds. Uemura explicitly leaves open whether the construction extends to other triangulations and arbitrary hyperbolic knots, and the general conjecture.

Substantial progress then came from the FAMED framework of Ben Aribi and Wong (2024, revised 2026). A FAMED ('Face Adjacency Matrices with Edge Duality') triangulation is an ordered ideal triangulation of a one-cusped 3-manifold with trivial second homology whose face-adjacency matrix $\mathcal A$ and Neumann–Zagier matrices $\mathbf A,\mathbf B$ — the matrices encoding Thurston's gluing equations for the shape parameters with respect to a chosen peripheral curve such as the knot longitude — satisfy explicit combinatorial linear-algebra conditions: non-emptiness of the space of angle structures, invertibility of $\mathcal A$ and $\mathbf B$, and an explicit matrix identity encoding the edge duality. These conditions alone force the algebraic simplifications in the partition function and the saddle-point structure needed for the asymptotic analysis, reducing the previously case-by-case analytic work to a finite combinatorial check. A geometric triangulation is one whose shapes realize the complete hyperbolic structure with positive dihedral angles. Wong (2025) further introduced generalized FAMED semi-geometric triangulations, relaxing the invertibility of $\mathcal A$ (allowing it to have nonzero nullity) and allowing semi-geometric triangulations, in which every tetrahedron is either geometric or flat (degenerate, of zero volume). Several aspects remain open: whether every hyperbolic knot complement admits a FAMED geometric triangulation or, more generally, a generalized FAMED semi-geometric triangulation; the H-triangulation clause (2); and the conjecture's quantification over all fully balanced shaped ideal triangulations. Establishing such a triangulation would resolve the reformulated clauses (1) and (3) only existentially in the triangulation.

Problem Statement

Prove or refute the Andersen–Kashaev volume conjecture in its remaining generality. As stated in the source (Conjecture 2.22 of arXiv:2307.12848, following Andersen–Kashaev 2014): for every oriented closed 3-manifold $M$ and every hyperbolic knot $K\subset M$, the conjecture asserts the existence of a smooth function $J_{M,K}(\hbar,x)$ on $\mathbb R_{>0}\times\mathbb R$ such that:

(1) for any fully balanced shaped ideal triangulation $X$ of the complementary space $M\setminus K$, there exist a real linear combination $\lambda$ of gauge-invariant dihedral angles and a real second-order polynomial $\phi$ of (not necessarily gauge-invariant) dihedral angles such that

$$Z_\hbar(X)=e^{i\phi/\hbar}\int_{\mathbb R}J_{M,K}(\hbar,x)\,e^{-x\lambda/\sqrt{\hbar}}\,dx;$$

(2) for any one-vertex shaped H-triangulation $Y$ of the pair $(M,K)$, there exists a real second-order polynomial $\varphi$ of dihedral angles such that

$$\lim_{\omega_Y\to\tau}\Phi_b\!\left(\frac{\pi-\omega_Y(K)}{2\pi i\sqrt\hbar}\right)Z_\hbar(Y)=e^{i\varphi/\hbar-i\pi/12}J_{M,K}(\hbar,0),$$

where $\omega_Y$ is the weight function of the shape structure on $Y$ (the sum of the dihedral angles around each edge), $\omega_Y(K)$ is its value on the edge representing $K$, $\Phi_b$ is Faddeev's quantum dilogarithm, and the limit is taken over shape structures of $Y$ as $\omega_Y$ tends to the function $\tau$ that takes the value $0$ on the edge representing $K$ and $2\pi$ on all other edges;

(3) the hyperbolic volume is recovered by

$$\lim_{\hbar\to 0^+}2\pi\hbar\,\log|J_{M,K}(\hbar,0)|=-\mathrm{Vol}(M\setminus K).$$

References

1. Soichiro Uemura, “A proof of the Teichmüller TQFT volume conjecture for $7_3$ knot”, arXiv:2307.12848 (2023), https://arxiv.org/abs/2307.12848
2. Jørgen Ellegaard Andersen and Rinat Kashaev, “A TQFT from Quantum Teichmüller Theory”, Communications in Mathematical Physics 330 (2014) 887–934, DOI 10.1007/s00220-014-2073-2, https://doi.org/10.1007/s00220-014-2073-2
3. Jørgen Ellegaard Andersen and Jens-Jakob Kratmann Nissen, “Asymptotic aspects of the Teichmüller TQFT”, arXiv:1612.06982 (2016), https://arxiv.org/abs/1612.06982
4. Fathi Ben Aribi, François Guéritaud and Eiichi Piguet-Nakazawa, “Geometric triangulations and the Teichmüller TQFT volume conjecture for twist knots”, Quantum Topology 14 (2023) 285–406, DOI 10.4171/qt/178, arXiv:1903.09480, https://doi.org/10.4171/qt/178
5. Fathi Ben Aribi and Ka Ho Wong, “The Andersen-Kashaev volume conjecture for FAMED geometric triangulations”, arXiv:2410.10776 (2024; revised February 2026), https://arxiv.org/abs/2410.10776
6. Fathi Ben Aribi, Antonin Guilloux and Ka Ho Wong, “FAMED by computer: proving the Andersen-Kashaev volume conjecture for 42,000 knots”, arXiv:2512.17437 (2025), https://arxiv.org/abs/2512.17437
7. Ka Ho Wong, “Asymptotics aspects of Teichmüller TQFT for generalized FAMED semi-geometric triangulations”, arXiv:2512.23198 (2025), https://arxiv.org/abs/2512.23198
8. Rinat Kashaev, “The Hyperbolic Volume of Knots from the Quantum Dilogarithm”, Letters in Mathematical Physics 39 (1997) 269–275, DOI 10.1023/A:1007364912784, https://doi.org/10.1023/A:1007364912784
9. Hitoshi Murakami and Jun Murakami, “The colored Jones polynomials and the simplicial volume of a knot”, Acta Mathematica 186 (2001) 85–104, DOI 10.1007/BF02392716, https://doi.org/10.1007/BF02392716
10. Eiichi Piguet-Nakazawa, “Teichmüller TQFT Calculations for Infinite Families of Knots”, PhD thesis, University of Geneva (2021), https://archive-ouverte.unige.ch/unige:152763