The Unbearable Hardness of Unknotting

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

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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


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)


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
  • Knot
  • Link
  • NP-hard
  • Reidemeister move
  • Unknot recognition
  • Unlinking number
  • intermediate invariants


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