Document Open Access Logo

The Unbearable Hardness of Unknotting

Authors Arnaud de Mesmay, Yo'av Rieck, Eric Sedgwick, Martin Tancer

PDF
Thumbnail PDF

File

LIPIcs.SoCG.2019.49.pdf
  • Filesize: 0.96 MB
  • 19 pages

Document Identifiers

Author Details

Arnaud de Mesmay
  • Univ. Grenoble Alpes, CNRS, Grenoble INP, GIPSA-lab, 38000 Grenoble, France
Yo'av Rieck
  • Department of Mathematical Sciences, University of Arkansas Fayetteville, AR 72701, USA
Eric Sedgwick
  • School of Computing, DePaul University, 243 S. Wabash Ave, Chicago, IL 60604, USA
Martin Tancer
  • Department of Applied Mathematics, Charles University, Malostranské nám. 25, 118 00 Praha 1, Czech Republic

Funding

  • de Mesmay, Arnaud: Partially supported by the French ANR projects ANR-16-CE40-0009-01 (GATO), the CNRS PEPS project COMP3D and the Czech-French collaboration project EMBEDS II (CZ: 7AMB17FR029, FR: 38087RM).
  • Rieck, Yo'av: This work was partially supported by a grant from the Simons Foundation (grant number 283495 to Yo'av Rieck).
  • Tancer, Martin: Partially supported by the Czech-French collaboration project EMBEDS II (CZ: 7AMB17FR029, FR: 38087RM), by the project CE-ITI (GAČR P202/12/G061) and by the Charles University projects PRIMUS/17/SCI/3 and UNCE/SCI/004.

Acknowledgements

We thank Amey Kaloti and Jeremy Van Horn Morris for many helpful conversations. We are grateful to the anonymous referees for a careful reading of the paper and many helpful suggestions.

Cite AsGet BibTex

Arnaud de Mesmay, Yo'av Rieck, Eric Sedgwick, and Martin Tancer. The Unbearable Hardness of Unknotting. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 49:1-49:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.SoCG.2019.49

Abstract

We prove that deciding if a diagram of the unknot can be untangled using at most k Reidemeister moves (where k is part of the input) is NP-hard. We also prove that several natural questions regarding links in the 3-sphere are NP-hard, including detecting whether a link contains a trivial sublink with n components, computing the unlinking number of a link, and computing a variety of link invariants related to four-dimensional topology (such as the 4-ball Euler characteristic, the slicing number, and the 4-dimensional clasp number).

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Geometric topology
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Knot
  • Link
  • NP-hard
  • Reidemeister move
  • Unknot recognition
  • Unlinking number
  • intermediate invariants

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Ian Agol, Joel Hass, and William Thurston. The computational complexity of knot genus and spanning area. Transactions of the American Mathematical Society, 358:3821-3850, 2006. URL: http://dx.doi.org/10.1090/S0002-9947-05-03919-X.
  2. Francesca Aicardi. Tree-like curves. Singularities and bifurcations, 21:1-31, 1994. Google Scholar
  3. Vladimir I Arnold. Plane curves, their invariants, perestroikas and classifications. Advances in Soviet Mathematics, 21:33-91, 1994. Google Scholar
  4. Vladimir Igorevich Arnold. Topological invariants of plane curves and caustics, volume 5. American Mathematical Soc., 1994. Google Scholar
  5. Sanjeev Arora and Boaz Barak. Computational complexity: a modern approach. Cambridge University Press, Cambridge, 2009. URL: http://dx.doi.org/10.1017/CBO9780511804090.
  6. Hsien-Chih Chang and Jeff Erickson. Untangling planar curves. Discrete &Computational Geometry, 58(4):889-920, 2017. URL: http://dx.doi.org/10.1007/s00454-017-9907-6.
  7. Arnaud de Mesmay, Yo'av Rieck, Eric Sedgwick, and Martin Tancer. Embeddability in ℝ³ is NP-hard. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1316-1329. Society for Industrial and Applied Mathematics, 2018. Full version on https://arxiv.org/abs/1604.00290. URL: http://dx.doi.org/10.1137/1.9781611975031.86.
  8. Arnaud de Mesmay, Yo'av Rieck, Eric Sedgwick, and Martin Tancer. The unbearable hardness of unknotting. https://arxiv.org/abs/1810.03502 (for numbering of claims, we refer to the version v1), 2018.
  9. Michael R. Garey and David S. Johnson. Computers and intractability. W. H. Freeman and Co., San Francisco, Calif., 1979. A guide to the theory of NP-completeness, A Series of Books in the Mathematical Sciences. Google Scholar
  10. Wolfgang Haken. Theorie der Normalflächen. Acta Mathematica, 105(3-4):245-375, 1961. Google Scholar
  11. Joel Hass and Jeffrey Lagarias. The number of Reidemeister moves needed for unknotting. Journal of the American Mathematical Society, 14(2):399-428, 2001. URL: http://dx.doi.org/10.1090/S0894-0347-01-00358-7.
  12. 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. URL: http://dx.doi.org/10.1145/301970.301971.
  13. Joel Hass and Tahl Nowik. Unknot diagrams requiring a quadratic number of Reidemeister moves to untangle. Discrete &Computational Geometry, 44(1):91-95, 2010. URL: http://dx.doi.org/10.1007/s00454-009-9156-4.
  14. Louis H Kauffman and Sofia Lambropoulou. Hard unknots and collapsing tangles. In Introductory Lectures on Knot Theory, 2014. URL: http://dx.doi.org/10.1142/9789814313001_0009.
  15. Dale Koenig and Anastasiia Tsvietkova. NP-hard problems naturally arising in knot theory, 2018. URL: http://arxiv.org/abs/1809.10334.
  16. Greg Kuperberg. Knottedness is in NP, modulo GRH. Adv. Math., 256:493-506, 2014. URL: http://dx.doi.org/10.1016/j.aim.2014.01.007.
  17. Greg Kuperberg and Eric Samperton. Coloring invariants of knots and links are often intractable. Manuscript. Google Scholar
  18. Marc Lackenby. A polynomial upper bound on Reidemeister moves. Ann. Math. (2), 182(2):491-564, 2015. URL: http://dx.doi.org/10.4007/annals.2015.182.2.3.
  19. Marc Lackenby. The efficient certification of knottedness and Thurston norm, 2016. URL: http://arxiv.org/abs/1604.00290.
  20. Marc Lackenby. Elementary knot theory. In Lectures on Geometry (Clay Lecture Notes). Oxford University Press, 2017. Google Scholar
  21. Marc Lackenby. Some conditionally hard problems on links and 3-manifolds. Discrete &Computational Geometry, 58(3):580-595, 2017. URL: http://dx.doi.org/10.1007/s00454-017-9905-8.
  22. Adam Simon Levine. Slicing mixed Bing-Whitehead doubles. Journal of Topology, 5(3):713-726, 2012. URL: http://dx.doi.org/10.1112/jtopol/jts019.
  23. 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
  24. Eric Samperton. Computational Complexity of Enumerative 3-Manifold Invariants. arXiv preprint, 2018. URL: http://arxiv.org/abs/1805.09275.
  25. Tetsuo Shibuya. Some relations among various numerical invariants for links. Osaka J. Math., 11:313-322, 1974. URL: http://0-projecteuclid.org.library.uark.edu/euclid.ojm/1200757391.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact