For a finite-volume hyperbolic 3-manifold (a quotient of hyperbolic 3-space by a discrete group of isometries, closed or cusped), the volume is a topological invariant by Mostow-Prasad rigidity and a computable real number, so certified numerical computation can always prove two volumes different whenever they differ. No algorithm is known, however, that can prove two volumes equal. Schmalian (2025) identified this gap as the exact obstruction to converting his algorithms' finite lists, which are guaranteed to contain all minimal-volume hyperbolic link complements in a given 3-manifold, into exact lists. The underlying question is: does there exist an algorithm which, given two finite-volume hyperbolic 3-manifolds presented by triangulations, decides whether their hyperbolic volumes are equal? Since each volume is a finite sum of Bloch-Wigner dilogarithm values at algebraic arguments, the question sits at the interface of computational 3-manifold topology, the Bloch group and its regulator, and transcendence theory. No decision procedure and no undecidability result are known; the problem is open.

Background

A hyperbolic 3-manifold is a quotient of hyperbolic 3-space H3\mathbb{H}^3 by a torsion-free discrete group of isometries; it has finite volume exactly when it is closed or consists of a compact part together with finitely many cusp ends, each diffeomorphic to a torus times a half-line. Knot and link complements carrying complete finite-volume hyperbolic structures are the central examples: the complement of a hyperbolic link in a compact 3-manifold is a finite-volume cusped hyperbolic 3-manifold. By the Mostow-Prasad rigidity theorem, a finite-volume hyperbolic 3-manifold is determined up to isometry by its fundamental group, so its hyperbolic volume vol⁡(M)\operatorname{vol}(M) is a topological invariant.

Such manifolds are specified to computers by finite triangulations, and there are algorithms that decide whether a given triangulated 3-manifold is hyperbolic and, if so, produce a geometric (ideal) triangulation of it (Kuperberg; recorded as Theorem 3.8 in Schmalian's paper). From an ideal triangulation, one whose vertices sit at the cusps, the complete hyperbolic structure is found by solving Thurston's gluing equations, a system of polynomial equations with integer coefficients, for the shape parameter zj∈C∖{0,1}z_j\in\mathbb{C}\setminus\{0,1\} of each ideal tetrahedron. The volume is then a finite sum of values of the Bloch-Wigner dilogarithm D(z)=Im⁡(Li⁡2(z))+arg⁡(1−z)log⁡∣z∣D(z)=\operatorname{Im}(\operatorname{Li}_2(z))+\arg(1-z)\log|z|, the single-valued real-analytic cousin of the dilogarithm Li⁡2(z)=∑n≥1zn/n2\operatorname{Li}_2(z)=\sum_{n\geq 1}z^n/n^2 (Garoufalidis-Zagier):

vol⁡(M)=∑jD(zj).\operatorname{vol}(M)=\sum_j D(z_j).

The shape parameters of the complete structure are algebraic numbers, and exact-arithmetic software (Snap; Coulson-Goodman-Hodgson-Neumann) computes the hyperbolic structure to high precision, recovers an exact algebraic description of it, and verifies it. Hence each volume is a computable real number: computing two volumes with certified enclosures to sufficient precision eventually proves them different whenever they differ. Certified approximation cannot, however, prove two volumes equal.

The obstruction is number-theoretic. Volume sums as above lie in the image of the regulator on the Bloch group B(F)B(F) of the trace field FF of the manifold: B(F)B(F) is the quotient of the free abelian group on F∖{0,1}F\setminus\{0,1\} by the five-term relation of the dilogarithm; Suslin proved that B(F)B(F) is isomorphic, up to torsion, to the algebraic KK-group K3(F)K_3(F), with DD realizing the Borel regulator (Neumann's extended Bloch group paper; see the survey of Garoufalidis-Zagier). Deciding vol⁡(M)=vol⁡(N)\operatorname{vol}(M)=\operatorname{vol}(N) therefore means deciding that a certain finite sum of Bloch-Wigner values at algebraic arguments vanishes; beyond the five-term relation, no unconditional description of all rational linear relations among such values is known, so there is at present no candidate general method for certifying equality.

Two further facts delineate the question. First, by Jorgensen-Thurston theory the set of volumes of hyperbolic 3-manifolds is a countable well-ordered subset of R>0\mathbb{R}_{>0}, and only finitely many orientable hyperbolic 3-manifolds share any given volume (Hodgson-Masai): volume is a finite-to-one but not injective invariant. Volume coincidences genuinely occur, and every known one is certified by an ad hoc argument: for instance Ruberman's theorem that mutant knots in S3S^3, obtained by cutting the complement along a Conway sphere, a 2-sphere meeting the knot transversely in four points, and regluing by the half-turn rotation about the axis through two of the intersection points, have complements of equal volume; the equality vol⁡(m003)=vol⁡(m004)\operatorname{vol}(m003)=\operatorname{vol}(m004) between the figure-eight knot complement m004m004 and its sister manifold m003m003, which are related by volume-preserving mutation along the fibre of their structure as once-punctured torus bundles, that is, manifolds fibering over the circle with fibre a torus with one point removed (Kegel-Ray-Spreer-Thompson-Tillmann); and the census coincidences vol⁡(m006)=vol⁡(m007)\operatorname{vol}(m006)=\operatorname{vol}(m007) and vol⁡(m015)=vol⁡(m016)=vol⁡(m017)\operatorname{vol}(m015)=\operatorname{vol}(m016)=\operatorname{vol}(m017), also proved by Kegel-Ray-Spreer-Thompson-Tillmann by exhibiting equal tetrahedron shapes through the gluing and completeness equations (the mm-names are those of the standard SnapPy census of low-complexity cusped hyperbolic 3-manifolds). Second, the homeomorphism problem, equivalently for hyperbolic manifolds the isometry problem, for triangulated 3-manifolds is decidable (Kuperberg), but equality of volume is strictly weaker than isometry, so this decidability result does not settle the volume question.

The question arises concretely in classification programs. Schmalian proved that, given a compact orientable 3-manifold MM and k≥0k\geq 0, one can compute to any desired precision the minimal volume of a kk-component hyperbolic link in MM, and can output a finite list of link complements containing all minimisers. The one ingredient missing for exact lists of minimal-volume links is precisely a decision procedure for equality of hyperbolic volumes: candidates whose volumes are numerically indistinguishable cannot, in general, be compared exactly.

Problem Statement

Does there exist an algorithm which, given any two orientable finite-volume hyperbolic 3-manifolds MM and NN, each presented by a finite triangulation, halts after finitely many steps and correctly decides whether vol⁡(M)=vol⁡(N)\operatorname{vol}(M)=\operatorname{vol}(N)?

Any equivalent finite presentation of the inputs is acceptable: recognising hyperbolicity and producing a geometric (ideal) triangulation from an arbitrary triangulation is itself algorithmic, so the promise that the inputs are hyperbolic is not an additional assumption. The question is posed at the source's generality: all orientable finite-volume hyperbolic 3-manifolds, closed or cusped, with no restriction on the pair; the source's conventions (Section 1.1) take all 3-manifolds to be oriented and hyperbolic manifolds to be complete and of finite volume.

A resolution would in particular settle, for every compact orientable 3-manifold MM and k≥0k\geq 0, whether the finite lists output by the minimal-volume link algorithms can be pruned to exact lists of minimal-volume kk-component hyperbolic link complements in MM.

References

  1. Misha Schmalian, An algorithm for finding minimal volume hyperbolic links and the Dehn parental test, 2025, arXiv:2503.05671 (v1, 7 March 2025), https://arxiv.org/abs/2503.05671 (source of the open problem)
  2. Marc Kegel, Arunima Ray, Jonathan Spreer, Em Thompson, Stephan Tillmann, On a volume invariant of 3-manifolds, Expositiones Mathematicae 44(4) (2026), article 125782, DOI 10.1016/j.exmath.2026.125782, https://doi.org/10.1016/j.exmath.2026.125782 (preprint arXiv:2402.04839, https://arxiv.org/abs/2402.04839)
  3. Walter D. Neumann, Don Zagier, Volumes of hyperbolic three-manifolds, Topology 24(3) (1985) 307-332, DOI 10.1016/0040-9383(85)90004-7, https://doi.org/10.1016/0040-9383(85)90004-7
  4. Daniel Ruberman, Mutation and volumes of knots in S^3, Inventiones Mathematicae 90(1) (1987) 189-215, DOI 10.1007/BF01389038, https://doi.org/10.1007/BF01389038
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  6. Craig Hodgson, Hidetoshi Masai, On the number of hyperbolic 3-manifolds of a given volume, in Geometry and Topology Down Under, Contemporary Mathematics 597, American Mathematical Society, 2013, pages 295-320, DOI 10.1090/conm/597/11767, https://doi.org/10.1090/conm/597/11767 (preprint arXiv:1203.6551, https://arxiv.org/abs/1203.6551)
  7. David Coulson, Oliver A. Goodman, Craig D. Hodgson, Walter D. Neumann, Computing arithmetic invariants of 3-manifolds, Experimental Mathematics 9(1) (2000) 127-152, DOI 10.1080/10586458.2000.10504641, https://doi.org/10.1080/10586458.2000.10504641
  8. Walter D. Neumann, Extended Bloch group and the Cheeger-Chern-Simons class, Geometry & Topology 8 (2004) 413-474, DOI 10.2140/gt.2004.8.413, https://doi.org/10.2140/gt.2004.8.413
  9. Stavros Garoufalidis, Don Zagier, Hyperbolic 3-manifolds, the Bloch group, and the work of Walter Neumann, 2023, arXiv:2304.12545 (preprint), https://arxiv.org/abs/2304.12545
  10. BoGwang Jeon, Sunul Oh, Hyperbolic Dehn filling, volume, and transcendentality, arXiv:2308.11574 (v2, 2025; v1, 2023, by Sunul Oh alone), https://arxiv.org/abs/2308.11574
  11. Robert C. Haraway III, Practical bounds for a Dehn parental test, arXiv:1504.01674 (v1 2015, v4 2018), https://arxiv.org/abs/1504.01674