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A Practical Algorithm for Knot Factorisation

Authors Alexander He , Eric Sedgwick , Jonathan Spreer

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ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Mathematics of computing → Combinatorial algorithms
  • Mathematics of computing → Geometric topology
Keywords
  • Prime and composite knots
  • (crushing) normal surfaces
  • edge-ideal triangulations
  • co-NP certificate
  • triangulation complexity

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Abstract

We present an algorithm for computing the prime factorisation of a knot, which is practical in the following sense: using Regina, we give an implementation that works well for inputs of reasonable size, including prime knots from the 19-crossing census. The main new ingredient in this work is an object that we call an "edge-ideal triangulation", which is what our algorithm uses to represent knots. As other applications, we give an alternative proof that prime knot recognition is in coNP, and present some new complexity results for triangulations. Beyond knots, our work showcases edge-ideal triangulations as a tool for potential applications in 3-manifold topology.

Cite As Get BibTex

Alexander He, Eric Sedgwick, and Jonathan Spreer. A Practical Algorithm for Knot Factorisation. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 55:1-55:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.SoCG.2025.55

Author Details

Alexander He
  • 401 Mathematical Sciences, Oklahoma State University, Stillwater, OK, USA
Eric Sedgwick
  • School of Computing, DePaul University, Chicago, IL, USA
Jonathan Spreer
  • School of Mathematics and Statistics, The University of Sydney, Australia

Funding

  • He, Alexander: Supported by an Australian Government Research Training Program Scholarship for part of this project.
  • Spreer, Jonathan: Supported by the Australian Research Council under the Discovery Project scheme (grant number DP220102588).

Acknowledgements

This work was inspired by discussions at the Dagstuhl workshop "Triangulations in Geometry and Topology" [M. Buchin et al., 2024]. The authors would like to thank the participants for interesting conversations about knots and normal surfaces. Moreover, the authors thank Nathan Dunfield for suggesting a way to generate hard diagrams of composite knots on which to test our algorithm. We also thank the anonymous reviewers for their helpful comments.

Supplementary Materials

References

  1. I. Agol, J. Hass, and W. Thurston. The computational complexity of knot genus and spanning area. Trans. Amer. Math. Soc., 358(9):3821-3850, 2006. URL: https://doi.org/10.1090/S0002-9947-05-03919-X.
  2. E. G. Altmann and J. Spreer. Sampling triangulations of manifolds using Monte Carlo methods. Exp. Math., 0(0):1-15, 2025. URL: https://doi.org/10.1080/10586458.2024.2433506.
  3. F. B. Aribi, F. Guéritaud, and E. Piguet-Nakazawa. Geometric triangulations and the teichmüller tqft volume conjecture for twist knots. Quantum Topol., 14(2):285-406, 2023. Google Scholar
  4. J. A. Baldwin and S. Sivek. On the complexity of torus knot recognition. Trans. Amer. Math. Soc., 371:3831-3855, 2019. URL: https://doi.org/10.1090/tran/7394.
  5. B. Benedetti. Discrete Morse theory for manifolds with boundary. Trans. Amer. Math. Soc., 364(12):6631-6670, 2012. URL: https://doi.org/10.1090/S0002-9947-2012-05614-5.
  6. B. Benedetti and F. H. Lutz. Knots in collapsible and non-collapsible balls. Electron. J. Combin., 20(3):Paper 31, 29, 2013. URL: https://doi.org/10.37236/3319.
  7. M. Buchin, J. Cardinal, A. de Mesmay, J. Spreer, and A. He. Triangulations in geometry and topology (dagstuhl seminar 24072). Dagstuhl Reports, 14(2):120-163, 2024. URL: https://doi.org/10.4230/DAGREP.14.2.120.
  8. B. A. Burton. Converting between quadrilateral and standard solution sets in normal surface theory. Alg. Geom. Topol., 9:2121-2174, 2009. URL: https://doi.org/10.2140/agt.2009.9.2121.
  9. B. 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.
  10. B. A. Burton. A new approach to crushing 3-manifold triangulations. Discrete Comput. Geom., 52(1):116-139, 2014. URL: https://doi.org/10.1007/s00454-014-9572-y.
  11. B. A. Burton. The Next 350 Million Knots. In Sergio Cabello and Danny Z. Chen, editors, 36th International Symposium on Computational Geometry (SoCG 2020), volume 164 of Leibniz International Proceedings in Informatics (LIPIcs), pages 25:1-25:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.SoCG.2020.25.
  12. B. A. Burton, R. Budney, W. Pettersson, et al. Regina: Software for low-dimensional topology, 1999-2023. Version 7.3. URL: https://regina-normal.github.io.
  13. B. A. Burton, T. de Paiva, A. He, and C. O. Y. Hui. Crushing surfaces of positive genus. arXiv:2403.11523, 2024. To appear in Algebr. Geom. Topol.. Google Scholar
  14. B. A. Burton and A. He. Finding large counterexamples by selectively exploring the Pachner graph. In Erin W. Chambers and Joachim Gudmundsson, editors, 39th International Symposium on Computational Geometry (SoCG 2023), volume 258 of Leibniz International Proceedings in Informatics (LIPIcs), pages 21:1-21:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.SoCG.2023.21.
  15. B. A. Burton and M. Özlen. A fast branching algorithm for unknot recognition with experimental polynomial-time behaviour. arXiv:1211.1079, 2012. To appear in Math. Program. Google Scholar
  16. N. M. Dunfield, M. Obeidin, and C. G. Rudd. Computing a Link Diagram from Its Exterior. In Xavier Goaoc and Michael Kerber, editors, 38th International Symposium on Computational Geometry (SoCG 2022), volume 224 of Leibniz International Proceedings in Informatics (LIPIcs), pages 37:1-37:24, Dagstuhl, Germany, 2022. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.SoCG.2022.37.
  17. J. Hass, J. C. Lagarias, and N. Pippenger. The computational complexity of knot and link problems. J. ACM, 46(2):185-211, 1999. URL: https://doi.org/10.1145/301970.301971.
  18. A. He, E. Sedgwick, and J. Spreer. A practical algorithm for knot factorisation. arXiv:2504.03942, 2025. Google Scholar
  19. Alexander He. AlexHe98/idealedge. Software, (visited on 2025-06-02). URL: https://github.com/AlexHe98/idealedge
    Software Heritage Logo archived version
    full metadata available at: https://doi.org/10.4230/artifacts.23286
  20. D. Ibarra, D. V. Mathews, J. S. Purcell, and J. Spreer. Triangulations of the 3-sphere with knotted edge. arXiv:2411.18938, 2024. Google Scholar
  21. K. Ichihara, Y. Nishimura, and S. Tani. The computational complexity of classical knot recognition. Journal of Knot Theory and Its Ramifications, 32(11):2350069, 2023. URL: https://doi.org/10.1142/S0218216523500694.
  22. A. Jackson. Recognition of Seifert fibered spaces with boundary is in NP. Math. Ann., 2024. URL: https://doi.org/10.1007/s00208-024-02920-x.
  23. W. Jaco and J. H. Rubinstein. 0-efficient triangulations of 3-manifolds. J. Differential Geom., 65(1):61-168, 2003. URL: https://doi.org/10.4310/jdg/1090503053.
  24. W. Jaco and J. L. Tollefson. Algorithms for the complete decomposition of a closed 3-manifold. Illinois J. Math., 39(3):358-406, 1995. Google Scholar
  25. G. Kuperberg. Knottedness is in NP, modulo GRH. Adv. Math., 256:493-506, 2014. URL: https://doi.org/10.1016/j.aim.2014.01.007.
  26. M. Lackenby. The efficient certification of knottedness and Thurston norm. Adv. Math., 387, 2021. Article ID: 107796. URL: https://doi.org/10.1016/j.aim.2021.107796.
  27. M. Lackenby. Unknot recognition in quasi-polynomial time. https://www.maths.ox.ac.uk/node/38304, 2021.
  28. W. B. R. Lickorish. Unshellable triangulations of spheres. European J. Combin., 12(6):527-530, 1991. URL: https://doi.org/10.1016/S0195-6698(13)80103-5.
  29. S. Schleimer. Sphere recognition lies in NP. In Low-dimensional and symplectic topology, volume 82 of Proc. Sympos. Pure Math., pages 183-213. Amer. Math. Soc., Providence, RI, 2011. URL: https://doi.org/10.1090/pspum/082/2768660.
  30. H. Schubert. Die eindeutige Zerlegbarkeit eines Knotens in Primknoten. Sitzungsberichte der Heidelberger Akademie der Wissenschaften. Springer, 1949. URL: https://doi.org/10.1007/978-3-642-45813-2.
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