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The Complete 10-Tetrahedra Census of Orientable Cusped Hyperbolic 3-Manifolds

Author Shana Yunsheng Li

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Geometric topology
Keywords
  • hyperbolic manifolds
  • 3-manifolds
  • triangulation
  • census
  • tabulation
  • exact computation
  • computational topology
  • low-dimensional topology

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Abstract

We extend the complete census of orientable cusped hyperbolic 3-manifolds to 10 tetrahedra, giving the next 150,730 manifolds and their 496,638 minimal ideal triangulations. As applications, we find the precisely 439,898 exceptional Dehn fillings on them, revealing the next 1,849 simplest hyperbolic knot exteriors in S³. We also give the simplest example of an orientable cusped hyperbolic 3-manifold containing a closed totally geodesic surface.

Cite As Get BibTex

Shana Yunsheng Li. The Complete 10-Tetrahedra Census of Orientable Cusped Hyperbolic 3-Manifolds. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 73:1-73:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SoCG.2026.73

Author Details

Shana Yunsheng Li
  • Dept. of Math., University of Illinois Urbana-Champaign, IL, USA

Funding

This material is based upon work supported by the National Science Foundation under Grant No. DMS-2424139 while the author was in residence at the Simons Laufer Mathematical Sciences Institute in Berkeley, California, during the Spring 2026 semester. The author was also supported by the National Science Foundation under Grant No. DMS-2303572 during the same semester.

Acknowledgements

The author wishes to thank Nathan Dunfield for his generous guidance during numerous discussions, and the Keeling cluster maintenance team at UIUC for providing the computational resources that made this project possible.

Supplementary Materials

References

  1. Ian Agol. Bounds on exceptional Dehn filling. Geom. Topol., 4(1):431-449, 2000. URL: https://doi.org/10.2140/gt.2000.4.431.
  2. Brannon Basilio, Chaeryn Lee, and Joseph Malionek. Totally Geodesic Surfaces in Hyperbolic 3-Manifolds: Algorithms and Examples. In Wolfgang Mulzer and Jeff M. Phillips, editors, 40th International Symposium on Computational Geometry (SoCG 2024), volume 293 of Leibniz International Proceedings in Informatics (LIPIcs), pages 14:1-14:19, Dagstuhl, Germany, 2024. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.SoCG.2024.14.
  3. Steven Boyer, Cameron McA Gordon, and Liam Watson. On L-spaces and left-orderable fundamental groups. Math. Ann., 356(4):1213-1245, 2013. URL: https://doi.org/10.1007/s00208-012-0852-7.
  4. Benjamin A. Burton. The Pachner graph and the simplification of 3-sphere triangulations. In Computational geometry (SCG'11), pages 153-162. ACM, New York, 2011. URL: https://doi.org/10.1145/1998196.1998220.
  5. Benjamin A. Burton. Computational topology with Regina: algorithms, heuristics and implementations. In Geometry and topology down under, volume 597 of Contemp. Math., pages 195-224. Amer. Math. Soc., Providence, RI, 2013. URL: https://doi.org/10.1090/conm/597/11877.
  6. Benjamin A. Burton. A duplicate pair in the SnapPea census. Exp. Math., 23(2):170-173, 2014. URL: https://doi.org/10.1080/10586458.2014.886535.
  7. Benjamin A. Burton. The cusped hyperbolic census is complete. Trans. Am. Math. Soc., pages 1-, 2017. URL: https://doi.org/10.1090/tran/6767.
  8. Benjamin A. Burton. The next 350 million knots. In 36th International Symposium on Computational Geometry, volume 164 of LIPIcs. Leibniz Int. Proc. Inform., pages Art. No. 25, 17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.SoCG.2020.25.
  9. Danny Calegari and Nathan M. Dunfield. Laminations and groups of homeomorphisms of the circle. Invent. Math., 152(1):149-204, 2003. URL: https://doi.org/10.1007/s00222-002-0271-6.
  10. Patrick J. Callahan, John C. Dean, and Jeffrey R. Weeks. The simplest hyperbolic knots. J. Knot Theory Ramifications, 8(3):279-297, 1999. URL: https://doi.org/10.1142/S0218216599000195.
  11. Patrick J. Callahan, Martin V. Hildebrand, and Jeffrey R. Weeks. A census of cusped hyperbolic 3-manifolds. Math. Comp., 68(225):321-332, 1999. With microfiche supplement. URL: https://doi.org/10.1090/S0025-5718-99-01036-4.
  12. Abhijit Champanerkar, Ilya Kofman, and Eric Patterson. The next simplest hyperbolic knots. J. Knot Theory Ramifications, 13(7):965-987, 2004. URL: https://doi.org/10.1142/S021821650400355X.
  13. Marc Culler, Nathan M. Dunfield, Matthias Goerner, and Jeffrey R. Weeks. SnapPy, a computer program for studying the geometry and topology of 3-manifolds. Available at http://snappy.computop.org (28/10/2025).
  14. Nathan Dunfield, Stavros Garoufalidis, and Seokbeom Yoon. Code and data for comparing 1-loop invariants and torsions, 2024. URL: https://doi.org/10.7910/DVN/XUJ3RH.
  15. Nathan M. Dunfield. Cyclic surgery, degrees of maps of character curves, and volume rigidity for hyperbolic manifolds. Invent. Math., 136(3):623-657, 1999. URL: https://doi.org/10.1007/s002220050321.
  16. Nathan M Dunfield. Floer homology, group orderability, and taut foliations of hyperbolic 3–manifolds. Geom. Topol., 24(4):2075-2125, 2020. URL: https://doi.org/10.2140/gt.2020.24.2075.
  17. Nathan M. Dunfield. A census of exceptional Dehn fillings. In Characters in low-dimensional topology, volume 760 of Contemp. Math., pages 143-155. Amer. Math. Soc., [Providence], RI, [2020] (C) 2020. URL: https://doi.org/10.1090/conm/760/15289.
  18. Nathan M. Dunfield, Stefan Friedl, and Nicholas Jackson. Twisted Alexander polynomials of hyperbolic knots. Exp. Math., 21(4):329-352, 2012. URL: https://doi.org/10.1080/10586458.2012.669268.
  19. Nathan M Dunfield, Stavros Garoufalidis, and Seokbeom Yoon. 1-loop equals torsion for fibered 3-manifolds. arXiv.org, 2024. URL: https://arxiv.org/abs/2304.00469.
  20. Nathan M. Dunfield, Neil R. Hoffman, and Joan E. Licata. Asymmetric hyperbolic L-spaces, Heegaard genus, and Dehn filling. Math. Res. Lett., 22(6):1679-1698, 2015. URL: https://doi.org/10.4310/MRL.2015.v22.n6.a7.
  21. D. B. A. Epstein and R. C. Penner. Euclidean decompositions of noncompact hyperbolic manifolds. J. Differential Geom., 27(1):67-80, 1988. URL: https://doi.org/10.4310/jdg/1214441650.
  22. Evgeny Fominykh, Stavros Garoufalidis, Matthias Goerner, Vladimir Tarkaev, and Andrei Vesnin. A census of tetrahedral hyperbolic manifolds. Exp. Math., 25(4):466-481, 2016. URL: https://doi.org/10.1080/10586458.2015.1114436.
  23. Giles Gardam. Profinite rigidity in the SnapPea census. Exp. Math., 30(4):489-498, 2021. URL: https://doi.org/10.1080/10586458.2019.1577766.
  24. Matthias Goerner. SnapPy: verified computations. Available at https://snappy.computop.org/verify.html (28/10/2025).
  25. Martin Hildebrand and Jeffrey Weeks. A computer generated census of cusped hyperbolic 3-manifolds. In Computers and mathematics (Cambridge, MA, 1989), pages 53-59. Springer, New York, 1989. URL: https://doi.org/10.1007/978-1-4613-9647-5_7.
  26. Craig D. Hodgson and Steven P. Kerckhoff. Universal bounds for hyperbolic Dehn surgery. Ann. Math., 162(1):367-421, 2005. URL: https://doi.org/10.4007/annals.2005.162.367.
  27. Neil Hoffman, Kazuhiro Ichihara, Masahide Kashiwagi, Hidetoshi Masai, Shin’ichi Oishi, and Akitoshi Takayasu. Verified computations for hyperbolic 3-manifolds. Exp. Math., 25(1):66-78, 2016. URL: https://doi.org/10.1080/10586458.2015.1029599.
  28. Jim Hoste, Morwen Thistlethwaite, and Jeff Weeks. The first 1,701,936 knots. Math. Intelligencer, 20(4):33-48, 1998. URL: https://doi.org/10.1007/BF03025227.
  29. William Jaco and Ulrich Oertel. An algorithm to decide if a 3-manifold is a Haken manifold. Topology, 23(2):195-209, 1984. URL: https://doi.org/10.1016/0040-9383(84)90039-9.
  30. András Juhász. A survey of Heegaard Floer homology. In New ideas in low dimensional topology, volume 56 of Ser. Knots Everything, pages 237-296. World Sci. Publ., Hackensack, NJ, 2015. URL: https://doi.org/10.1142/9789814630627_0007.
  31. Marc Lackenby. Word hyperbolic Dehn surgery. Invent. Math., 140(2):243-282, 2000. URL: https://doi.org/10.1007/s002220000047.
  32. A. K. Lenstra, H. W. Lenstra, Jr., and L. Lovász. Factoring polynomials with rational coefficients. Math. Ann., 261(4):515-534, 1982. URL: https://doi.org/10.1007/BF01457454.
  33. Shana Yunsheng Li. The complete 10-tetrahedra census of orientable cusped hyperbolic 3-manifolds, 2025. URL: https://arxiv.org/abs/2512.02142.
  34. Shana Yunsheng Li. The complete 10-tetrahedra census of orientable cusped hyperbolic 3-manifolds, 2025. URL: https://doi.org/10.7910/DVN/BFEXZA.
  35. Shana Yunsheng Li. snappy_11_tets, 2026. URL: https://pypi.org/project/snappy-11-tets/.
  36. Feng Luo, Saul Schleimer, and Stephan Tillmann. Geodesic ideal triangulations exist virtually. Proc. Amer. Math. Soc., 136(7):2625-2630, 2008. URL: https://doi.org/10.1090/S0002-9939-08-09387-8.
  37. Bruno Martelli. An introduction to geometric topology, 2022. URL: https://arxiv.org/abs/1610.02592.
  38. Peter Ozsváth and Zoltán Szabó. Holomorphic disks and knot invariants. Adv. Math., 186(1):58-116, 2004. URL: https://doi.org/10.1016/j.aim.2003.05.001.
  39. Stuart Rankin, Ortho Flint, and John Schermann. Enumerating the prime alternating knots, Part I. J. Knot Theory Ramifications, 13(1):57-100, 2004. URL: https://doi.org/10.1142/S0218216504003044.
  40. Makoto Sakuma and Jeffrey R. Weeks. The generalized tilt formula. Geom. Dedicata, 55(2):115-123, 1995. URL: https://doi.org/10.1007/BF01264924.
  41. Peter Guthrie Tait. Xix.—on knots. Part II. Transactions of the Royal Society of Edinburgh, 32(2):327-342, 1884. URL: https://doi.org/10.1017/S0080456800026806.
  42. Morwen Thistlethwaite. Index of /∼mthistle/8tet. Available at https://web.math.utk.edu/~mthistle/8tet/ (04/02/2026).
  43. Morwen B. Thistlethwaite. The enumeration and classification of prime 20-crossing knots. Algebr. Geom. Topol., 25(1):329-344, 2025. URL: https://doi.org/10.2140/agt.2025.25.329.
  44. Jeffrey R. Weeks. Convex hulls and isometries of cusped hyperbolic 3-manifolds. Topology Appl., 52(2):127-149, 1993. URL: https://doi.org/10.1016/0166-8641(93)90032-9.
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