Document Open Access Logo

On Sparse Representations of 3‑Manifolds

Authors Kristóf Huszár , Clément Maria

PDF

File

Thumbnail PDF
PDF
LIPIcs.SoCG.2026.58.pdf
  • Filesize: 2.09 MB
  • 18 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Geometric topology
  • Theory of computation → Fixed parameter tractability
Keywords
  • computational 3-manifold topology
  • fixed-parameter tractability
  • Heegaard splittings and diagrams
  • triangulations
  • edge valence
  • treewidth
  • quantum invariants
  • tensor networks

Metrics

Abstract

3-manifolds are commonly represented as triangulations, consisting of abstract tetrahedra whose triangular faces are identified in pairs. The combinatorial sparsity of a triangulation, as measured by the treewidth of its dual graph, plays a fundamental role in the design of parameterized algorithms. In this work, we investigate algorithmic procedures that transform or modify a given triangulation while controlling specific sparsity parameters. First, we revisit a standard, linear-time algorithm that converts a given triangulation into a Heegaard diagram of the underlying 3-manifold, showing that the construction preserves treewidth. We apply this construction to exhibit a fixed-parameter tractable framework for computing Kuperberg’s quantum invariants of 3-manifolds. Second, we present a quasi-linear-time algorithm that retriangulates a given triangulation into one with maximum edge valence of at most nine, while only moderately increasing the treewidth of the dual graph. Combining these two algorithms yields a quasi-linear-time algorithm that produces, from a given triangulation, a Heegaard diagram in which every attaching curve intersects at most nine others.

Cite As Get BibTex

Kristóf Huszár and Clément Maria. On Sparse Representations of 3‑Manifolds. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 58:1-58:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SoCG.2026.58

Author Details

Kristóf Huszár
  • Institute of Geometry, Graz University of Technology, Austria
Clément Maria
  • Inria d'Université Côte d'Azur, Sophia Antipolis, France

Funding

  • Maria, Clément: Partially supported by the ANR project ANR-20-CE48-0007 (AlgoKnot).

References

  1. D. Bienstock. On embedding graphs in trees. J. Comb. Theory, Ser. B, 49(1):103-136, 1990. URL: https://doi.org/10.1016/0095-8956(90)90066-9.
  2. N. Brady, J. McCammond, and J. Meier. Bounding edge degrees in triangulated 3-manifolds. Proc. Amer. Math. Soc., 132(1):291-298, 2004. URL: https://doi.org/10.1090/S0002-9939-03-06981-8.
  3. B. A. Burton. The HOMFLY-PT polynomial is fixed-parameter tractable. In 34th Int. Symp. Comput. Geom. (SoCG 2018), volume 99 of LIPIcs. Leibniz Int. Proc. Inform., pages 18:1-18:14. Schloss Dagstuhl-Leibniz-Zent. Inf., 2018. URL: https://doi.org/10.4230/LIPIcs.SoCG.2018.18.
  4. B. A. Burton, R. Budney, W. Pettersson, et al. Regina: Software for low-dimensional topology, 1999-2025. Version 7.4.1. URL: https://regina-normal.github.io.
  5. B. A. Burton and R. G. Downey. Courcelle’s theorem for triangulations. J. Comb. Theory, Ser. A, 146:264-294, 2017. URL: https://doi.org/10.1016/j.jcta.2016.10.001.
  6. B. A. Burton, T. Lewiner, J. Paixão, and J. Spreer. Parameterized complexity of discrete Morse theory. ACM Trans. Math. Softw., 42(1):6:1-6:24, 2016. URL: https://doi.org/10.1145/2738034.
  7. B. A. Burton, C. Maria, and J. Spreer. Algorithms and complexity for Turaev-Viro invariants. J. Appl. Comput. Topol., 2(1-2):33-53, 2018. URL: https://doi.org/10.1007/s41468-018-0016-2.
  8. B. A. Burton and W. Pettersson. Fixed parameter tractable algorithms in combinatorial topology. In Proc. 20th Int. Conf. Comput. Comb. (COCOON 2014), volume 8591 of Lect. Notes Comput. Sci., pages 300-311. Springer, 2014. URL: https://doi.org/10.1007/978-3-319-08783-2_26.
  9. B. A. Burton and J. Spreer. The complexity of detecting taut angle structures on triangulations. In Proc. 24th Annu. ACM-SIAM Symp. Discrete Algorithms (SODA 2013), pages 168-183, 2013. URL: https://doi.org/10.1137/1.9781611973105.13.
  10. C. Damm, M. Holzer, and P. McKenzie. The complexity of tensor calculus. Comput. Complexity, 11(1-2):54-89, 2002. URL: https://doi.org/10.1007/s00037-000-0170-4.
  11. A. de Mesmay, J. Purcell, S. Schleimer, and E. Sedgwick. On the tree-width of knot diagrams. J. Comput. Geom., 10(1):164-180, 2019. URL: https://doi.org/10.20382/jocg.v10i1a6.
  12. D. Eppstein. What is ... treewidth? Notices Amer. Math. Soc., 72(2):172-175, 2025. URL: https://www.ams.org/journals/notices/202502/rnoti-p172.pdf.
  13. F. Frick. Combinatorial restrictions on cell complexes. PhD thesis, TU Berlin, 2015. URL: https://doi.org/10.14279/depositonce-4545.
  14. A. He, J. Morgan, and E. K. Thompson. An algorithm to construct one-vertex triangulations of Heegaard splittings. J. Comput. Geom., 16(1):635-693, November 2025. URL: https://doi.org/10.20382/jocg.v16i1a18.
  15. P. Heegaard. Forstudier til en topologisk Teori for de algebraiske Fladers Sammenhæng (in Danish). PhD thesis, Det Nordiske Forlag, 1898. URL: https://books.google.com/books?id=p9hMAAAAYAAJ. An English translation is available at URL: https://webhomes.maths.ed.ac.uk/~v1ranick/papers/heegaardenglish.pdf.
  16. P. Heegaard. Sur l'"Analysis situs". Bull. Soc. Math. France, 44:161-242, 1916. URL: https://doi.org/10.24033/bsmf.968.
  17. K. Huszár. On the pathwidth of hyperbolic 3-manifolds. Comput. Geom. Topol., 1(1):1-19, 2022. URL: https://doi.org/10.57717/cgt.v1i1.4.
  18. K. Huszár and C. Maria. On sparse representations of 3-manifolds, 2025. URL: https://arxiv.org/abs/2512.05779.
  19. K. Huszár and J. Spreer. 3-Manifold triangulations with small treewidth. In 35th Int. Symp. Comput. Geom. (SoCG 2019), volume 129 of LIPIcs. Leibniz Int. Proc. Inf., pages 44:1-44:20. Schloss Dagstuhl-Leibniz-Zent. Inf., 2019. URL: https://doi.org/10.4230/LIPIcs.SoCG.2019.44.
  20. K. Huszár and J. Spreer. On the width of complicated JSJ decompositions. Discrete Comput. Geom., 74(4):917-943, 2025. URL: https://doi.org/10.1007/s00454-025-00746-1.
  21. K. Huszár, J. Spreer, and U. Wagner. On the treewidth of triangulated 3-manifolds. J. Comput. Geom., 10(2):70-98, 2019. URL: https://doi.org/10.20382/jogc.v10i2a5.
  22. G. Kuperberg. Involutory Hopf algebras and 3-manifold invariants, 1990. URL: https://arxiv.org/abs/math/9201301.
  23. G. Kuperberg. Involutory Hopf algebras and 3-manifold invariants. Internat. J. Math., 2(1):41-66, 1991. URL: https://doi.org/10.1142/S0129167X91000053.
  24. G. Kuperberg. Noninvolutory Hopf algebras and 3-manifold invariants. Duke Math. J., 84(1):83-129, 1996. URL: https://doi.org/10.1215/S0012-7094-96-08403-3.
  25. M. Lackenby. Algorithms in 3-manifold theory. In I. Agol and D. Gabai, editors, Surveys in 3-manifold topology and geometry, volume 25 of Surv. Differ. Geom., pages 163-213. Int. Press Boston, 2022. URL: https://doi.org/10.4310/SDG.2020.v25.n1.a5.
  26. C. Lunel and A. de Mesmay. A Structural Approach to Tree Decompositions of Knots and Spatial Graphs. In 39th Int. Symp. Comput. Geom. (SoCG 2023), volume 258 of LIPIcs. Leibniz Int. Proc. Inf., pages 50:1-50:16. Schloss Dagstuhl-Leibniz-Zent. Inf., 2023. URL: https://doi.org/10.4230/LIPIcs.SoCG.2023.50.
  27. J. A. Makowsky. Coloured Tutte polynomials and Kauffman brackets for graphs of bounded tree width. Discrete Appl. Math., 145(2):276-290, 2005. URL: https://doi.org/10.1016/j.dam.2004.01.016.
  28. J. A. Makowsky and J. P. Mariño. The parametrized complexity of knot polynomials. J. Comput. Syst. Sci., 67(4):742-756, 2003. Special issue on parameterized computation and complexity. URL: https://doi.org/10.1016/S0022-0000(03)00080-1.
  29. C. Maria. Parameterized complexity of quantum knot invariants. In 37th Int. Symp. Comput. Geom. (SoCG 2021), volume 189 of LIPIcs. Leibniz Int. Proc. Inf., pages 53:1-53:15. Schloss Dagstuhl - Leibniz-Zent. Inf., 2021. URL: https://doi.org/10.4230/LIPIcs.SoCG.2021.53.
  30. C. Maria and J. Purcell. Treewidth, crushing and hyperbolic volume. Algebr. Geom. Topol., 19(5):2625-2652, 2019. URL: https://doi.org/10.2140/agt.2019.19.2625.
  31. I. L. Markov and Y. Shi. Simulating quantum computation by contracting tensor networks. SIAM J. Comput., 38(3):963-981, 2008. URL: https://doi.org/10.1137/050644756.
  32. S. Matveev. Algorithmic Topology and Classification of 3-Manifolds, volume 9 of Algorithms Comput. Math. Springer, Berlin, 2nd edition, 2007. URL: https://doi.org/10.1007/978-3-540-45899-9.
  33. E. E. Moise. Affine structures in 3-manifolds. V. The triangulation theorem and Hauptvermutung. Ann. Math. (2), 56:96-114, 1952. URL: https://doi.org/10.2307/1969769.
  34. B. O'Gorman. Parameterization of tensor network contraction. In 14th Conf. Theory Quantum Comput. Commun. Cryptogr., volume 135 of LIPIcs. Leibniz Int. Proc. Inform., pages 10:1-10:19. Schloss Dagstuhl. Leibniz-Zent. Inform., 2019. URL: https://doi.org/10.4230/LIPIcs.TQC.2019.10.
  35. T. Radó. Über den Begriff der Riemannschen Fläche. Acta Sci. Math., 2(2):101-121, 1925. Google Scholar
  36. N. Robertson and P. D. Seymour. Graph minors. II. Algorithmic aspects of tree-width. J. Algorithms, 7(3):309-322, 1986. URL: https://doi.org/10.1016/0196-6774(86)90023-4.
  37. J. H. Rubinstein, H. Segerman, and S. Tillmann. Traversing three-manifold triangulations and spines. Enseign. Math., 65(1-2):155-206, 2019. URL: https://doi.org/10.4171/lem/65-1/2-5.
  38. M. Scharlemann. Heegaard splittings of compact 3-manifolds. In Handbook of Geometric Topology, pages 921-953. North-Holland, 2001. URL: https://doi.org/10.1016/B978-044482432-5/50019-6.
  39. J. Schultens. Introduction to 3-Manifolds, volume 151 of Grad. Stud. Math. Am. Math. Soc., Providence, RI, 2014. URL: https://doi.org/10.1090/gsm/151.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2026 Schloss Dagstuhl – LZI GmbH Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact