Document Open Access Logo

A Structural Approach to Tree Decompositions of Knots and Spatial Graphs

Authors Corentin Lunel, Arnaud de Mesmay

Thumbnail PDF


  • Filesize: 0.67 MB
  • 16 pages

Document Identifiers

Author Details

Corentin Lunel
  • LIGM, CNRS, Univ. Gustave Eiffel, ESIEE Paris, F-77454 Marne-la-Vallée, France
Arnaud de Mesmay
  • LIGM, CNRS, Univ. Gustave Eiffel, ESIEE Paris, F-77454 Marne-la-Vallée, France


We would like to thank Pierre Dehornoy and Saul Schleimer for helpful discussions, and the anonymous reviewers for their feedback which allowed us to significantly improve the paper.

Cite AsGet BibTex

Corentin Lunel and Arnaud de Mesmay. A Structural Approach to Tree Decompositions of Knots and Spatial Graphs. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 50:1-50:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


Knots are commonly represented and manipulated via diagrams, which are decorated planar graphs. When such a knot diagram has low treewidth, parameterized graph algorithms can be leveraged to ensure the fast computation of many invariants and properties of the knot. It was recently proved that there exist knots which do not admit any diagram of low treewidth, and the proof relied on intricate low-dimensional topology techniques. In this work, we initiate a thorough investigation of tree decompositions of knot diagrams (or more generally, diagrams of spatial graphs) using ideas from structural graph theory. We define an obstruction on spatial embeddings that forbids low tree width diagrams, and we prove that it is optimal with respect to a related width invariant. We then show the existence of this obstruction for knots of high representativity, which include for example torus knots, providing a new and self-contained proof that those do not admit diagrams of low treewidth. This last step is inspired by a result of Pardon on knot distortion.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Mathematics of computing → Geometric topology
  • Knots
  • Spatial Graphs
  • Tree Decompositions
  • Tangle
  • Representativity


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


  1. Colin C. Adams. The knot book. American Mathematical Society, 1994. Google Scholar
  2. Omid Amini, Frédéric Mazoit, Nicolas Nisse, and Stéphan Thomassé. Submodular partition functions. Discrete Mathematics, 309(20):6000-6008, 2009. Google Scholar
  3. Rudolph H Bing. The geometric topology of 3-manifolds, volume 40. American Mathematical Society, 1983. Google Scholar
  4. Ryan Blair and Makoto Ozawa. Height, trunk and representativity of knots. Journal of the Mathematical Society of Japan, 71(4):1105-1121, 2019. Google Scholar
  5. Hans L. Bodlaender, Benjamin A. Burton, Fedor V. Fomin, and Alexander Grigoriev. Knot diagrams of treewidth two. In International Workshop on Graph-Theoretic Concepts in Computer Science, pages 80-91. Springer, 2020. Google Scholar
  6. Benjamin A. Burton. The HOMFLY-PT polynomial is fixed-parameter tractable. In 34th International Symposium on Computational Geometry (SoCG 2018). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2018. Google Scholar
  7. Benjamin A. Burton, Herbert Edelsbrunner, Jeff Erickson, and Stephan Tillmann. Computational geometric and algebraic topology. Oberwolfach Reports, 12(4):2637-2699, 2016. Google Scholar
  8. Benjamin A. Burton, Clément Maria, and Jonathan Spreer. Algorithms and complexity for TuraevendashViro invariants. Journal of Applied and Computational Topology, 2(1-2):33-53, 2018. Google Scholar
  9. Arnaud de Mesmay, Jessica Purcell, Saul Schleimer, and Eric Sedgwick. On the tree-width of knot diagrams. Journal of Computational Geometry, 10(1):164-180, 2019. Google Scholar
  10. Reinhard Diestel. Graph theory. Number 173 in Graduate texts in mathematics. Springer, New York, 5th edition, 2016. Google Scholar
  11. Reinhard Diestel and Sang-il Oum. Tangle-Tree Duality: In Graphs, Matroids and Beyond. Combinatorica, 39(4):879-910, August 2019. Google Scholar
  12. Reinhard Diestel and Sang-il Oum. Tangle-tree duality in abstract separation systems. Advances in Mathematics, 377:107470, 2021. Google Scholar
  13. Reinhard Diestel and Geoff Whittle. Tangles and the Mona Lisa. arXiv preprint arXiv:1603.06652, 2016. Google Scholar
  14. Frederic Dorn, Eelko Penninkx, Hans L Bodlaender, and Fedor V Fomin. Efficient exact algorithms on planar graphs: Exploiting sphere cut branch decompositions. In European Symposium on Algorithms, pages 95-106. Springer, 2005. Google Scholar
  15. David B.A. Epstein. Curves on 2-manifolds and isotopies. Acta Mathematica, 115:83-107, 1966. Google Scholar
  16. David Gabai. Foliations and the topology of 3-manifolds. ii. Journal of Differential Geometry, 26(3):461-478, 1987. Google Scholar
  17. James Geelen, Bert Gerards, Neil Robertson, and Geoff Whittle. Obstructions to branch-decomposition of matroids. Journal of Combinatorial Theory, page 11, 2006. Google Scholar
  18. Martin Grohe. Tangles and connectivity in graphs. In Language and Automata Theory and Applications: 10th International Conference, LATA 2016, Prague, Czech Republic, March 14-18, 2016, Proceedings 10, pages 24-41. Springer, 2016. Google Scholar
  19. Mikhael Gromov. Filling Riemannian manifolds. Journal of Differential Geometry, 18(1):1-147, 1983. Google Scholar
  20. Misha Gromov and Larry Guth. Generalizations of the Kolmogorov-Barzdin embedding estimates. Duke Mathematical Journal, 161(13):2549-2603, 2012. Google Scholar
  21. 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
  22. Allen Hatcher. Algebraic topology. Cambridge University Press, Cambridge ; New York, 2002. Google Scholar
  23. Chuichiro Hayashi and Koya Shimokawa. Thin position of a pair (3-manifold, 1-submanifold). Pacific Journal of Mathematics, 197(2):301-324, February 2001. Google Scholar
  24. Qidong He and Scott A Taylor. Links, bridge number, and width trees. Journal of the Mathematical Society of Japan, 1(1):1-39, 2022. Google Scholar
  25. Kristóf Huszár and Jonathan Spreer. 3-manifold triangulations with small treewidth. In 35th International Symposium on Computational Geometry (SoCG 2019). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2019. Google Scholar
  26. Kristóf Huszár, Jonathan Spreer, and Uli Wagner. On the treewidth of triangulated 3-manifolds. Journal of Computational Geometry, 10(2):70-98, 2019. Google Scholar
  27. François Jaeger, Dirk L. Vertigan, and Dominic J.A. Welsh. On the computational complexity of the Jones and Tutte polynomials. In Mathematical Proceedings of the Cambridge Philosophical Society, volume 108, pages 35-53. Cambridge University Press, 1990. Google Scholar
  28. Marc Lackenby. Algorithms in 3-manifold theory. Surveys in Differential Geometry, 2020. Google Scholar
  29. Marc Lackenby. The efficient certification of knottedness and Thurston norm. Advances in Mathematics, 387:107796, 2021. Google Scholar
  30. Marc Lackenby. Unknot recognition in quasi-polynomial time, 2021. Talk with slides available on the author’s webpage : quasipolynomial-talk.pdf. Google Scholar
  31. Corentin Lunel and Arnaud de Mesmay. A structural approach to tree decompositions of knots and spatial graphs. arXiv preprint arXiv:2303.07982, 2023. Google Scholar
  32. Laurent Lyaudet, Frédéric Mazoit, and Stéphan Thomassé. Partitions versus sets: a case of duality. European journal of Combinatorics, 31(3):681-687, 2010. Google Scholar
  33. J.A. Makowsky and J.P. Mariño. The parametrized complexity of knot polynomials. Journal of Computer and System Sciences, 67(4):742-756, December 2003. Google Scholar
  34. Clément Maria. Parameterized Complexity of Quantum Knot Invariants. In Kevin Buchin and Éric Colin de Verdière, editors, 37th International Symposium on Computational Geometry (SoCG 2021), volume 189 of Leibniz International Proceedings in Informatics (LIPIcs), pages 53:1-53:17, Dagstuhl, Germany, 2021. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. Google Scholar
  35. Clément Maria and Jessica Purcell. Treewidth, crushing and hyperbolic volume. Algebraic & Geometric Topology, 19(5):2625-2652, 2019. Google Scholar
  36. Makoto Ozawa. Bridge position and the representativity of spatial graphs. Topology and its Applications, 159(4):936-947, 2012. Google Scholar
  37. John Pardon. On the distortion of knots on embedded surfaces. Annals of Mathematics, 174(1):637-646, July 2011. Google Scholar
  38. Neil Robertson and Paul D. Seymour. Graph minors. V. Excluding a planar graph. Journal of Combinatorial Theory, Series B, 41(1):92-114, August 1986. Google Scholar
  39. Neil Robertson and Paul D. Seymour. Graph minors. X. Obstructions to tree-decomposition. Journal of Combinatorial Theory, Series B, 52(2):153-190, July 1991. Google Scholar
  40. Neil Robertson and Paul D. Seymour. Graph minors. XI. Circuits on a Surface. Journal of Combinatorial Theory. Series B, 60(1):72-106, January 1994. Google Scholar
  41. Neil Robertson and Paul D. Seymour. Graph minors. XX. Wagner’s conjecture. Journal of Combinatorial Theory, Series B, 92(2):325-357, 2004. Google Scholar
  42. Horst Sachs. On a spatial analogue of Kuratowski’s theorem on planar graphs—an open problem. In Graph theory, pages 230-241. Springer, 1983. Google Scholar
  43. Martin Scharlemann. Thin position in the theory of classical knots. In Handbook of knot theory, pages 429-459. Elsevier, 2005. Google Scholar
  44. Martin Scharlemann, Jennifer Schultens, and Toshio Saito. Lecture notes on generalized Heegaard splittings. World Scientific, 2016. Google Scholar
  45. Martin Scharlemann and Abigail Thompson. Thin position for 3-manifolds. In Geometric Topology: Joint US-Israel Workshop on Geometric Topology, June 10-16, 1992, Technion, Haifa, Israel, volume 164, page 231. American Mathematical Society, 1994. Google Scholar
  46. Jennifer Schultens. Introduction to 3-manifolds, volume 151. American Mathematical Society, 2014. Google Scholar
  47. Jennifer Schultens. The bridge number of a knot. In Encyclopedia of Knot Theory, pages 229-242. Chapman and Hall/CRC, 2021. Google Scholar
  48. Paul D. Seymour and Robin Thomas. Call routing and the ratcatcher. Combinatorica, 14(2):217-241, 1994. Google Scholar
  49. Scott Taylor and Maggy Tomova. Additive invariants for knots, links and graphs in 3-manifolds. Geometry & Topology, 22(6):3235-3286, 2018. Google Scholar
  50. Alan Mathison Turing. Solvable and unsolvable problems. Penguin Books London, 1954. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail