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The Unbearable Hardness of Unknotting

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

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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

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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).

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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

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.

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