Document Open Access Logo

Computing a Link Diagram from Its Exterior

Authors Nathan M. Dunfield , Malik Obeidin, Cameron Gates Rudd

Thumbnail PDF


  • Filesize: 3.47 MB
  • 24 pages

Document Identifiers

Author Details

Nathan M. Dunfield
  • Dept. of Math., University of Illinois at Urbana-Champaign, IL, USA
Malik Obeidin
  • Google, Inc., Mountain View, CA, USA
Cameron Gates Rudd
  • Dept. of Math., University of Illinois at Urbana-Champaign, IL, USA


We thank Matthias Goerner and Henry Segerman for helpful correspondence, and thank the referees for their detailed comments which helped improve this paper.

Cite AsGet BibTex

Nathan M. Dunfield, Malik Obeidin, and Cameron Gates Rudd. Computing a Link Diagram from Its Exterior. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 37:1-37:24, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


A knot is a circle piecewise-linearly embedded into the 3-sphere. The topology of a knot is intimately related to that of its exterior, which is the complement of an open regular neighborhood of the knot. Knots are typically encoded by planar diagrams, whereas their exteriors, which are compact 3-manifolds with torus boundary, are encoded by triangulations. Here, we give the first practical algorithm for finding a diagram of a knot given a triangulation of its exterior. Our method applies to links as well as knots, and allows us to recover links with hundreds of crossings. We use it to find the first diagrams known for 23 principal congruence arithmetic link exteriors; the largest has over 2,500 crossings. Other applications include finding pairs of knots with the same 0-surgery, which relates to questions about slice knots and the smooth 4D Poincaré conjecture.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Geometric topology
  • computational topology
  • low-dimensional topology
  • knot
  • knot exterior
  • knot diagram
  • link
  • link exterior
  • link diagram


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


  1. Colin Adams. Triple crossing number of knots and links. J. Knot Theory Ramifications, 22(2):1350006, 17, 2013. URL:
  2. Colin C. Adams. Isometric cusps in hyperbolic 3-manifolds. Michigan Math. J., 46(3):515-531, 1999. URL:
  3. Colin Conrad. Adams. The knot book : an elementary introduction to the mathematical theory of knots. W.H. Freeman, 1994. Google Scholar
  4. Kenneth L. Baker. A sketchy surgery description of the seifert-weber dodecahedral space, 2021. URL:
  5. Kenneth L. Baker and Marc Kegel. Census L-space knots are braid positive, except one that is not, in preparation. Google Scholar
  6. M. D. Baker, M. Goerner, and A. W. Reid. All principal congruence link groups. J. Algebra, 528:497-504, 2019. URL:
  7. Mark D. Baker, Matthias Goerner, and Alan W. Reid. All known principal congruence links. Preprint 2019, 9 pages. URL:
  8. Benjamin A. Burton. The cusped hyperbolic census is complete. Preprint 2014, 32 pages. URL:
  9. 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:
  10. 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:
  11. 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-Zent. Inform., Wadern, 2020. URL:
  12. Benjamin A. Burton, Hsien-Chih Chang, Maarten Löffler, Arnaud de Mesmay, Clément Maria, Saul Schleimer, Eric Sedgwick, and Jonathan Spreer. Hard diagrams of the unknot. Preprint 2021, 26 pages. URL:
  13. Benjamin A. Burton, J. Hyam Rubinstein, and Stephan Tillmann. The Weber-Seifert dodecahedral space is non-Haken. Trans. Amer. Math. Soc., 364(2):911-932, 2012. URL:
  14. Abhijit Champanerkar, Ilya Kofman, and Timothy Mullen. The 500 simplest hyperbolic knots. J. Knot Theory Ramifications, 23(12):1450055, 34, 2014. URL:
  15. Marc Culler, Nathan M. Dunfield, Matthias Goerner, and Jeffrey R. Weeks. SnapPy, a computer program for studying the geometry and topology of 3-manifolds, version 3.0.2, 2021. URL:
  16. Arnaud de Mesmay, Yo'av Rieck, Eric Sedgwick, and Martin Tancer. The unbearable hardness of unknotting. Adv. Math., 381:Paper No. 107648, 36, 2021. URL:
  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. URL:
  18. Nathan M. Dunfield, Malik Obeidin, and Cameron Gates Rudd. Computing a Link Diagram from its Exterior, 2021. Full version of this paper, 34 pages. URL:
  19. Nathan M. Dunfield, Malik Obeidin, and Cameron Gates Rudd. Code and data for computing a link diagram from its exterior, 2022. URL:
  20. I. A. Dynnikov. Three-page approach to knot theory. Coding and local motions. Funktsional. Anal. i Prilozhen., 33(4):25-37, 96, 1999. URL:
  21. I. A. Dynnikov. Arc-presentations of links: monotonic simplification. Fund. Math., 190:29-76, 2006. URL:
  22. Ivan Dynnikov and Vera Sokolova. Multiflypes of rectangular diagrams of links. J. Knot Theory Ramifications, 30(6):Paper No. 2150038, 15, 2021. URL:
  23. Erica Flapan. When topology meets chemistry. Outlooks. Cambridge University Press, Cambridge; Mathematical Association of America, Washington, DC, 2000. A topological look at molecular chirality. URL:
  24. Erica Flapan, Adam He, and Helen Wong. Topological descriptions of protein folding. Proc. Natl. Acad. Sci. USA, 116(19):9360-9369, 2019. URL:
  25. Michael Freedman, Robert Gompf, Scott Morrison, and Kevin Walker. Man and machine thinking about the smooth 4-dimensional Poincaré conjecture. Quantum Topol., 1(2):171-208, 2010. URL:
  26. C. McA. Gordon and J. Luecke. Knots are determined by their complements. J. Amer. Math. Soc., 2(2):371-415, 1989. URL:
  27. Wolfgang Haken. Theorie der Normalflächen. Acta Math., 105:245-375, 1961. URL:
  28. Joel Hass, Jeffrey C. Lagarias, and Nicholas Pippenger. The computational complexity of knot and link problems. J. ACM, 46(2):185-211, 1999. URL:
  29. Craig D. Hodgson and Jeffrey R. Weeks. Symmetries, isometries and length spectra of closed hyperbolic three-manifolds. Experiment. Math., 3(4):261-274, 1994. Google Scholar
  30. Jim Hoste, Morwen Thistlethwaite, and Jeff Weeks. The first 1,701,936 knots. Math. Intelligencer, 20(4):33-48, 1998. URL:
  31. S. V. Ivanov. The computational complexity of basic decision problems in 3-dimensional topology. Geom. Dedicata, 131:1-26, 2008. URL:
  32. William Jaco and J. Hyam Rubinstein. Inflations of ideal triangulations. Adv. Math., 267:176-224, 2014. URL:
  33. William Jaco and Eric Sedgwick. Decision problems in the space of Dehn fillings. Topology, 42(4):845-906, 2003. URL:
  34. Francesco Lin and Michael Lipnowski. Monopole Floer Homology, Eigenform Multiplicities and the Seifert-Weber Ddodecahedral Space. Int. Math. Res. Notices, to appear. URL:
  35. Ciprian Manolescu and Lisa Piccirillo. From zero surgeries to candidates for exotic definite four-manifolds. Preprint 2021, 30 pages. URL:
  36. Sergei Matveev. Algorithmic topology and classification of 3-manifolds, volume 9 of Algorithms and Computation in Mathematics. Springer, Berlin, second edition, 2007. Google Scholar
  37. Aleksandar Mijatović. Simplifying triangulations of S³. Pacific J. Math., 208(2):291-324, 2003. URL:
  38. Malik Obeidin. Link simplification code for Spherogram. URL:
  39. Udo Pachner. P.L. homeomorphic manifolds are equivalent by elementary shellings. European J. Combin., 12(2):129-145, 1991. URL:
  40. Satya R. T. Peddada, Nathan M. Dunfield, Lawrence E. Zeidner, Kai A. James, and James T. Allison. Systematic Enumeration and Identification of Unique Spatial Topologies of 3D Systems Using Spatial Graph Representations. In International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, volume 3A: 47th Design Automation Conference (DAC), 2021. URL:
  41. Lisa Piccirillo. The Conway knot is not slice. Ann. of Math. (2), 191(2):581-591, 2020. URL:
  42. Riccardo Piergallini. Standard moves for standard polyhedra and spines. Rend. Circ. Mat. Palermo (2) Suppl., 18:391-414, 1988. Third National Conference on Topology (Italian) (Trieste, 1986). Google Scholar
  43. Dale Rolfsen. Knots and links, volume 7 of Mathematics Lecture Series. Publish or Perish, Inc., Houston, TX, 1990. Corrected reprint of the 1976 original. Google Scholar
  44. Stefan Schirra. Robustness and precision issues in geometric computation. In Handbook of computational geometry, pages 597-632. North-Holland, Amsterdam, 2000. URL:
  45. Saul 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:
  46. Henry Segerman. Connectivity of triangulations without degree one edges under 2-3 and 3-2 moves. Proc. Amer. Math. Soc., 145(12):5391-5404, 2017. URL:
  47. Carl Sundberg and Morwen Thistlethwaite. The rate of growth of the number of prime alternating links and tangles. Pacific J. Math., 182(2):329-358, 1998. URL:
  48. The PARI Group, Univ. Bordeaux. PARI/GP version 2.11.4, 2020. URL:
  49. Stephan Tillmann. Normal surfaces in topologically finite 3-manifolds. Enseign. Math. (2), 54(3-4):329-380, 2008. URL:
  50. C. Weber and H. Seifert. Die beiden Dodekaederräume. Math. Z., 37(1):237-253, 1933. URL:
  51. Jeff Weeks. Computation of hyperbolic structures in knot theory. In Handbook of knot theory, pages 461-480. Elsevier B. V., Amsterdam, 2005. URL:
  52. Jeffery R. Weeks. Source code file close_cusp.c for SnapPea, version 2.5, circa 1995. URL:
  53. Jeffrey R. Weeks. Convex hulls and isometries of cusped hyperbolic 3-manifolds. Topology Appl., 52(2):127-149, 1993. URL:
  54. Raphael Zentner. Integer homology 3-spheres admit irreducible representations in SL(2,ℂ). Duke Math. J., 167(9):1643-1712, 2018. URL:
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