Universality of the generalized FAMED condition for ideal triangulations of hyperbolic knot complements (Conjectures 1.2 and 1.8 of Wong, 2025)

The Andersen-Kashaev volume conjecture predicts that the hyperbolic volume of a knot complement can be recovered from the semi-classical limit of the Teichmüller TQFT knot invariant. Recent work of Wong (arXiv:2512.23198) proves an asymptotic expansion of the Teichmüller TQFT partition function and the Jones function, and hence the Andersen-Kashaev volume conjecture, for any hyperbolic knot whose complement admits a semi-geometric triangulation satisfying a purely combinatorial 'generalized FAMED' condition on its face adjacency and Neumann-Zagier matrices. What remains open, and is posed here, is whether this combinatorial condition is universal: (Conjecture 1.8) does every hyperbolic knot complement in S^3 admit a semi-geometric triangulation that is generalized FAMED with respect to (longitude, meridian), and (Conjecture 1.2, the universal form) is every ordered ideal triangulation with a nonempty space of angle structures generalized FAMED with respect to the longitude? An affirmative answer to Conjecture 1.8 would, through the already-proved Corollary 1.16, settle the Andersen-Kashaev volume conjecture for all hyperbolic knots in S^3. Existing evidence is computational only (more than 42,000 knot complements verified for the original FAMED condition), and no proof or counterexample is known.

Background

A hyperbolic knot $K \subset S^3$ has a complement $S^3 \smallsetminus K$ carrying a complete finite-volume hyperbolic metric. Such a manifold can be decomposed into ideal tetrahedra (tetrahedra whose vertices lie at the cusp and are removed); an ideal triangulation $X$ with $N$ tetrahedra is a combinatorial encoding of this decomposition, and the hyperbolic structure is described by shape parameters $z_k \in \mathbb{C}$ satisfying the gluing equations: around each edge of the triangulation, the product of incident shapes is 1. An angle structure on $X$ is an assignment of positive angles to the three pairs of opposite edges of each tetrahedron (summing to $\pi$ per tetrahedron) such that the angles around each edge of the triangulation sum to $2\pi$; the space $\mathscr{A}_X$ of angle structures is a convex polytope which is nonempty for many but not all triangulations. The triangulation is ordered (branched) if each tetrahedron carries a vertex order respected by the face pairings, which fixes a sign $\varepsilon(T) = \pm 1$ per tetrahedron. Wong (2025) builds from this data the face adjacency matrices: for $k = 0,\dots,3$, the integer matrix $\mathscr{X}_k \in M_{N,2N}(\mathbb{Z})$ whose $(i,j)$ entry is 1 when the $j$-th face of the triangulation is the face of the $i$-th tetrahedron opposite its $k$-th vertex and 0 otherwise; the diagonal sign matrix $\mathscr{E} \in M_{N \times N}(\mathbb{Z})$; and the vertical block combinations $\mathscr{A} = (\mathscr{X}_0 - \mathscr{X}_1 + \mathscr{X}_2; \mathscr{X}_2 - \mathscr{X}_3) \in M_{2N \times 2N}(\mathbb{Z})$ and $\mathscr{B} = (0_N; \mathscr{E}) \in M_{2N \times N}(\mathbb{Z})$.

Independently, the gluing equations together with a holonomy equation for the preferred longitude $l$ (the peripheral curve, unique up to sign, that is null-homologous in the complement) can be written in logarithmic shape parameters using the Neumann-Zagier matrices $G, G', G'' \in M_{N \times N}(\mathbb{Z})$ (Neumann-Zagier 1985); setting $A := G - G'$ and $B := G'' - G'$, the system reads $A\,\mathrm{Log}\,\mathbf{z} + B\,\mathrm{Log}\,\mathbf{z}'' = i\nu + \tilde{u}$ with $\nu \in \pi\mathbb{Z}^N$ and $\tilde{u} = (0,\dots,0,\xi)^\top$ encoding the longitude holonomy $\xi$. Write $n := \mathrm{nullity}(\mathscr{A})$ for the nullity of the face adjacency matrix $\mathscr{A}$ (this is the nullity entering the source's conditions — not that of the Neumann-Zagier matrix $A$); the source's standing hypothesis is $\mathrm{nullity}(B) = 2n = 2\,\mathrm{nullity}(\mathscr{A})$, and $E \in M_{N \times N}(\mathbb{Q})$ is then a product of elementary row operations putting the block matrix $(EB|EA)$ into reduced row echelon form, decomposed into blocks $(EB)_{N-2n}, (EB)_{2n}$ and $(EA)_{N-2n}, (EA)_{2n}$.

The Teichmüller TQFT of Andersen and Kashaev (2014) is a topological quantum field theory constructed from quantum Teichmüller theory (the quantization of the Teichmüller spaces of punctured surfaces); to an ideally triangulated 3-manifold with angle structure it assigns a partition function $\mathscr{Z}_\hbar(X,\alpha)$, defined via Faddeev's quantum dilogarithm, depending on a parameter $\hbar > 0$. The Andersen-Kashaev volume conjecture (their Conjecture 1(1)&(3)) predicts, for every hyperbolic knot, a Jones function $\mathfrak{J}_X$ — conjecturally identified with the Kashaev invariant of the knot, an $R$-matrix state sum expected to coincide with a specialization of the colored Jones polynomial — whose integral against an exponential reproduces $|\mathscr{Z}_\hbar|$, and whose semi-classical limit obeys $\lim_{\hbar \to 0^+} 2\pi\hbar \log|\mathfrak{J}_X(\hbar,0)| = -\mathrm{Vol}(S^3 \smallsetminus K)$. Prior to Wong (2025), the conjecture was verified only for special families: the knot $6_1$ (Andersen-Nissen), all twist knots (Ben Aribi-Guéritaud-Piguet-Nakazawa), and the knot $7_3$ (Uemura).

The decisive new input is combinatorial. Ben Aribi and Wong (2024) introduced FAMED triangulations ('Face Adjacency Matrices with Edge Duality'): a linear-algebraic condition on $\mathscr{A}, \mathscr{B}, \mathscr{E}$ under which the saddle-point analysis of $\mathscr{Z}_\hbar$ can be carried out, and they proved the Andersen-Kashaev conjecture for all FAMED geometric triangulations. Here 'geometric' means every tetrahedron has a positively oriented shape (whether every cusped hyperbolic 3-manifold admits a geometric triangulation is itself a well-known open problem; however, triangulating the Epstein-Penner polyhedral decomposition and inserting flat tetrahedra — degenerate tetrahedra with shape $1$ — always yields a semi-geometric triangulation, one whose tetrahedra are geometric or flat). Since some ordered ideal triangulations are known not to be FAMED, Wong (2025) defined the generalized FAMED condition (Definition 1.1): $X$ is generalized FAMED with respect to $l$ if (1) the angle structure space $\mathscr{A}_X$ is nonempty, (2) $\mathrm{nullity}(B) = 2\,\mathrm{nullity}(\mathscr{A}) < N$, and (3)-(4) two explicit pairs of block matrices — built from the reduced blocks $(EA)_{2n}$ and $(EB)_{N-2n}$, the products $\mathscr{X}_0 E_2$ and $E_1 \mathscr{B}$ (where $E_1, E_2$ are the elementary matrices putting $\mathscr{A}$ into rank normal form), and the matrix $(\mathscr{E} + \mathrm{Id}_N)/2$ — are pairwise row equivalent over $\mathbb{Q}$. When $n = 0$ these conditions reduce exactly to the original FAMED condition. Definition 1.5 adds, for the pair $(l,m)$ with $m$ the meridian, two further conditions: (2) all entries of the last column of the block $E_{2n}$ of the reduced matrix $E$ are zero (this removes the longitude-holonomy parameter $\xi$ from the reduced gluing equations), and (3) the meridian $m$, oriented so that the algebraic intersection number of $l$ and $m$ is 1, admits a representative whose logarithmic holonomy is a fixed $\mathbb{Q}$-linear combination of the logarithms of the shape parameters at the pivot columns of $(EB)_{N-2n}$, plus a rational multiple of $\pi i$. The source notes that ordered triangulations which are generalized FAMED with respect to $l$ but fail condition (2) of Definition 1.5 do exist.

Wong's main theorems show that for any generalized FAMED semi-geometric triangulation, $\mathscr{Z}_\hbar(X,\alpha)$ has a full asymptotic expansion in $\hbar \to 0^+$ with exponential decay rate governed by the volume of a hyperbolic cone structure determined by $\alpha$, and the 1-loop invariant of Dimofte-Garoufalidis (a determinant-like function on the gluing variety, defined from the Neumann-Zagier datum in 'The quantum content of the gluing equations') appears as the 1-loop term; with Definition 1.5 satisfied, the Jones function exists and Corollary 1.16 deduces the Andersen-Kashaev volume conjecture for the knot. All of this is conditional on the combinatorial hypotheses, whose universality the paper can only conjecture: every step of the asymptotic analysis is done, and the sole gap between the current state and the volume conjecture for arbitrary hyperbolic knots is whether suitable triangulations always exist.

Problem Statement

Prove or disprove each of the two combinatorial conjectures posed in Wong, 'Asymptotics aspects of Teichmüller TQFT for generalized FAMED semi-geometric triangulations' (arXiv:2512.23198):

(1) Conjecture 1.8 (existence form). Every hyperbolic knot complement in $S^3$ admits a semi-geometric ideal triangulation (every tetrahedron geometric or flat — of positively oriented shape, or the degenerate shape $1$) that is generalized FAMED with respect to $(l,m)$ — i.e. satisfying Definition 1.5 of the source: generalized FAMED with respect to the preferred longitude $l$ in the sense of Definition 1.1 (nonempty angle structure space; $\mathrm{nullity}(B) = 2\,\mathrm{nullity}(\mathscr{A}) < N$ for the face adjacency matrix $\mathscr{A}$ and Neumann-Zagier matrix $B$; and two explicit row-equivalence-over-$\mathbb{Q}$ conditions on the block matrices built from $(EB)_{N-2n}$, $(EA)_{2n}$, $\mathscr{X}_0$, $\mathscr{E}$ and $\mathscr{B}$), together with the two additional conditions of Definition 1.5(2)-(3): all entries of the last column of the block $E_{2n}$ of the reduced row echelon matrix are zero, and the meridian $m$ (oriented so that the algebraic intersection number of $l$ and $m$ is 1) admits a representative whose logarithmic holonomy is an explicit $\mathbb{Q}$-linear combination of the shape-parameter logarithms at the pivot columns of $(EB)_{N-2n}$, up to a rational multiple of $\pi i$.

(2) Conjecture 1.2 (universal form). Every ordered ideal triangulation of a hyperbolic knot complement in $S^3$ that admits angle structures is generalized FAMED with respect to the preferred longitude $l$.

Refuting Conjecture 1.2 means exhibiting an explicit ordered ideal triangulation of a hyperbolic knot complement in $S^3$ with nonempty angle structure space that provably fails Definition 1.1, and refuting Conjecture 1.8 means exhibiting a hyperbolic knot complement in $S^3$ proven to admit no semi-geometric triangulation satisfying Definition 1.5. By Corollary 1.16 of the source (already proved), an affirmative answer to Conjecture 1.8, combined with Theorem 1.14, settles the Andersen-Kashaev volume conjecture for all hyperbolic knots in $S^3$; the problem posed here is this derived combinatorial question raised by the source, not the Andersen-Kashaev conjecture itself, and the Dimofte-Garoufalidis 1-loop conjecture (Conjecture 2.20 of the source) remains a separate open problem outside the scope of this question.

References

1. Ka Ho Wong, 'Asymptotics aspects of Teichmüller TQFT for generalized FAMED semi-geometric triangulations', arXiv:2512.23198 (2025), https://arxiv.org/abs/2512.23198
2. Fathi Ben Aribi, Antonin Guilloux, 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
3. Fathi Ben Aribi, Ka Ho Wong, 'The Andersen-Kashaev volume conjecture for FAMED geometric triangulations', arXiv:2410.10776 (2024), https://arxiv.org/abs/2410.10776
4. Jørgen Ellegaard Andersen, Rinat Kashaev, 'A TQFT from quantum Teichmüller theory', Communications in Mathematical Physics 330 (2014), no. 3, 887-934, doi:10.1007/s00220-014-2073-2, arXiv:1109.6295, https://arxiv.org/abs/1109.6295
5. Tudor D. Dimofte, Stavros Garoufalidis, 'The quantum content of the gluing equations', Geometry & Topology 17 (2013), no. 3, 1253-1315, doi:10.2140/gt.2013.17.1253, arXiv:1202.6268, https://arxiv.org/abs/1202.6268
6. Fathi Ben Aribi, François Guéritaud, Eiichi Piguet-Nakazawa, 'Geometric triangulations and the Teichmüller TQFT volume conjecture for twist knots', Quantum Topology 14 (2023), no. 2, 285-406, doi:10.4171/qt/178, arXiv:1903.09480, https://arxiv.org/abs/1903.09480
7. Tushar Pandey, Ka Ho Wong, 'Geometry of fundamental shadow link complements and applications to the 1-loop conjecture', arXiv:2308.06643 (2023), https://arxiv.org/abs/2308.06643
8. Seokbeom Yoon, 'The twisted 1-loop invariant and the Jacobian of Ptolemy varieties', Mathematische Zeitschrift 307 (2024), no. 1, article 19, doi:10.1007/s00209-024-03491-y; preprint arXiv:2110.11003, titled 'The twisted 1-loop invariant and the Jacobian of Ptolemy coordinates', https://arxiv.org/abs/2110.11003
9. Soichiro Uemura, 'A proof of the Teichmüller TQFT volume conjecture for 7_3 knot', arXiv:2307.12848 (2023), https://arxiv.org/abs/2307.12848
10. Jørgen Ellegaard Andersen, Jens-Jakob Kratmann Nissen, 'Asymptotic aspects of the Teichmüller TQFT', arXiv:1612.06982 (2016), to appear in Travaux Mathématiques, https://arxiv.org/abs/1612.06982