Algorithms for Contractibility of Compressed Curves on 3-Manifold Boundaries

Authors Erin Wolf Chambers, Francis Lazarus, Arnaud de Mesmay, Salman Parsa

Thumbnail PDF


  • Filesize: 0.75 MB
  • 16 pages

Document Identifiers

Author Details

Erin Wolf Chambers
  • St. Louis University, MO, USA
Francis Lazarus
  • G-SCOP, CNRS, UGA, Grenoble, France
Arnaud de Mesmay
  • LIGM, CNRS, Université Gustave Eiffel, ESIEE Paris, 77454 Marne-la-Vallée, France
Salman Parsa
  • St. Louis University, MO, USA


The authors would like to thank Mark Bell, Ben Burton, and Jeff Erickson for helpful discussions. We also thank an anonymous referee of [de Verdi{è}re and Parsa, 2020] for suggesting the use of a maximal compression body as a simple exponential-time algorithm for deciding contractibility of arbitrary (non-compressed) curves on the boundary of a 3-manifold.

Cite AsGet BibTex

Erin Wolf Chambers, Francis Lazarus, Arnaud de Mesmay, and Salman Parsa. Algorithms for Contractibility of Compressed Curves on 3-Manifold Boundaries. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 23:1-23:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


In this paper we prove that the problem of deciding contractibility of an arbitrary closed curve on the boundary of a 3-manifold is in NP. We emphasize that the manifold and the curve are both inputs to the problem. Moreover, our algorithm also works if the curve is given as a compressed word. Previously, such an algorithm was known for simple (non-compressed) curves, and, in very limited cases, for curves with self-intersections. Furthermore, our algorithm is fixed-parameter tractable in the complexity of the input 3-manifold. As part of our proof, we obtain new polynomial-time algorithms for compressed curves on surfaces, which we believe are of independent interest. We provide a polynomial-time algorithm which, given an orientable surface and a compressed loop on the surface, computes a canonical form for the loop as a compressed word. In particular, contractibility of compressed curves on surfaces can be decided in polynomial time; prior published work considered only constant genus surfaces. More generally, we solve the following normal subgroup membership problem in polynomial time: given an arbitrary orientable surface, a compressed closed curve γ, and a collection of disjoint normal curves Δ, there is a polynomial-time algorithm to decide if γ lies in the normal subgroup generated by components of Δ in the fundamental group of the surface after attaching the curves to a basepoint.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Geometric topology
  • Mathematics of computing → Graphs and surfaces
  • Theory of computation → Data compression
  • 3-manifolds
  • surfaces
  • low-dimensional topology
  • contractibility
  • compressed curves


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Ian Agol, Joel Hass, and William Thurston. The computational complexity of knot genus and spanning area. Transactions of the American Mathematical Society, 358(9):3821-3850, 2006. Google Scholar
  2. Matthias Aschenbrenner, Stefan Friedl, and Henry Wilton. Decision problems for 3-manifolds and their fundamental groups. Geometry & Topology Monographs, 19(1):201-236, 2015. Google Scholar
  3. Matthias Aschenbrenner, Stefan Friedl, Henry Wilton, and Stefan Friedl. 3-manifold groups, volume 20. European Mathematical Society Zürich, 2015. Google Scholar
  4. Mark C Bell. Simplifying triangulations. arXiv preprint, 2016. URL:
  5. Benjamin Burton, Éric Colin de Verdière, and Arnaud de Mesmay. On the complexity of immersed normal surfaces. Geometry & Topology, 20(2):1061-1083, 2016. Google Scholar
  6. Benjamin A Burton. A new approach to crushing 3-manifold triangulations. Discrete & Computational Geometry, 52(1):116-139, 2014. Google Scholar
  7. James W Cannon, David BA Epstein, Derek F Holt, Silvio VF Levy, Michael S Paterson, and William P Thurston. Word processing in groups. CRC Press, 1992. Google Scholar
  8. Pierre de La Harpe. Topics in geometric group theory. University of Chicago Press, 2000. Google Scholar
  9. Arnaud de Mesmay, Yo'av Rieck, Eric Sedgwick, and Martin Tancer. The Unbearable Hardness of Unknotting. In Gill Barequet and Yusu Wang, editors, 35th International Symposium on Computational Geometry (SoCG 2019), pages 49:1-49:19, Dagstuhl, Germany, 2019. Google Scholar
  10. Éric Colin de Verdière and Salman Parsa. Deciding contractibility of a non-simple curve on the boundary of a 3-manifold. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 2691-2704. SIAM, 2017. Google Scholar
  11. Éric Colin de Verdière and Salman Parsa. Deciding contractibility of a non-simple curve on the boundary of a 3-manifold: A computational loop theorem. arXiv preprint, 2020. URL:
  12. Max Dehn. Transformation der Kurven auf zweiseitigen Flächen. Mathematische Annalen, 72(3):413-421, 1912. Google Scholar
  13. Vincent Despré and Francis Lazarus. Computing the geometric intersection number of curves. Journal of the ACM (JACM), 66(6):1-49, 2019. Google Scholar
  14. Ivan Dynnikov. Counting intersections of normal curves. arXiv preprint, 2020. URL:
  15. Ivan Dynnikov and Bert Wiest. On the complexity of braids. Journal of the European Mathematical Society, 9:801-840, 2007. Google Scholar
  16. Jeff Erickson and Amir Nayyeri. Tracing compressed curves in triangulated surfaces. Discrete and Computational Geometry, 49:823-863, 2013. Google Scholar
  17. Jeff Erickson and Kim Whittlesey. Transforming curves on surfaces redux. In Proceedings of the twenty-fourth annual ACM-SIAM symposium on Discrete algorithms, pages 1646-1655. SIAM, 2013. Google Scholar
  18. Christian Hagenah. Gleichungen mit regulären Randbedingungen über freien Gruppen, 2000. Google Scholar
  19. Joel Hass, Jeffrey C Lagarias, and Nicholas Pippenger. The computational complexity of knot and link problems. Journal of the ACM (JACM), 46(2):185-211, 1999. Google Scholar
  20. Joel Hass, Jack Snoeyink, and William P Thurston. The size of spanning disks for polygonal curves. Discrete and Computational Geometry, 29(1):1-18, 2003. Google Scholar
  21. A. Hatcher, Cambridge University Press, and Cornell University. Department of Mathematics. Algebraic Topology. Algebraic Topology. Cambridge University Press, 2002. URL:
  22. John Hempel. 3-Manifolds, volume 349. American Mathematical Soc., 2004. Google Scholar
  23. Derek Holt, Markus Lohrey, and Saul Schleimer. Compressed Decision Problems in Hyperbolic Groups. In Rolf Niedermeier and Christophe Paul, editors, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019), volume 126, pages 37:1-37:16, Dagstuhl, Germany, 2019. Google Scholar
  24. William Jaco and Jeffrey L Tollefson. Algorithms for the complete decomposition of a closed 3-manifold. Illinois journal of mathematics, 39(3):358-406, 1995. Google Scholar
  25. Greg Kuperberg. Algorithmic homeomorphism of 3-manifolds as a corollary of geometrization. Pacific Journal of Mathematics, 301(1):189-241, 2019. Google Scholar
  26. Marc Lackenby. The efficient certification of knottedness and Thurston norm. arXiv preprint, 2016. URL:
  27. Marc Lackenby. Some conditionally hard problems on links and 3-manifolds. Discrete & Computational Geometry, 58(3):580-595, 2017. Google Scholar
  28. Marc Lackenby. Algorithms in 3-manifold theory. arXiv preprint, 2020. URL:
  29. Francis Lazarus and Julien Rivaud. On the homotopy test on surfaces. In 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science, pages 440-449. IEEE, 2012. Google Scholar
  30. Maarten Löffler, Anna Lubiw, Saul Schleimer, and Erin Moriarty Wolf Chambers. Computation in Low-Dimensional Geometry and Topology (Dagstuhl Seminar 19352). In Dagstuhl Reports, volume 9. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2019. Google Scholar
  31. Markus Lohrey. Word problems and membership problems on compressed words. SIAM Journal on Computing, 35(5):1210-1240, 2006. Google Scholar
  32. Markus Lohrey. The compressed word problem for groups. Springer, 2014. Google Scholar
  33. Bojan Mohar and Carsten Thomassen. Graphs on Surfaces. John Hopkins University Press, 2001. Google Scholar
  34. Edwin E Moise. Affine structures in 3-manifolds: V. The Triangulation theorem and Hauptvermutung. Annals of mathematics, pages 96-114, 1952. Google Scholar
  35. Marcus Schaefer, Eric Sedgwick, and Daniel Štefankovič. Algorithms for normal curves and surfaces. In International Computing and Combinatorics Conference, pages 370-380. Springer, 2002. Google Scholar
  36. Marcus Schaefer, Eric Sedgwick, and Daniel Stefankovic. Computing Dehn twists and geometric intersection numbers in polynomial time. In CCCG, volume 20, pages 111-114. Citeseer, 2008. Google Scholar
  37. Saul Schleimer. Polynomial-time word problems. Commentarii Mathematici Helvetici, 83:741-765, 2008. Google Scholar
  38. Horst Schubert. Bestimmung der Primfaktorzerlegung von Verkettungen. Mathematische Zeitschrift, 76(1):116-148, 1961. Google Scholar
  39. John Stillwell. Classical topology and combinatorial group theory, volume 72. Springer Science & Business Media, 2012. Google Scholar
  40. Friedhelm Waldhausen. The word problem in fundamental groups of sufficiently large irreducible 3-manifolds. Annals of Mathematics, 88(2):272-280, 1968. Google Scholar