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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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On the complexity of torus knot recognition
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by John A. Baldwin and Steven Sivek PDF
Trans. Amer. Math. Soc. 371 (2019), 3831-3855 Request permission


We show that the problem of recognizing that a knot diagram represents a specific torus knot, or any torus knot at all, is in the complexity class NP${}\cap {}$co-NP, assuming the generalized Riemann hypothesis. We also show that satellite knot detection is in NP under the same assumption and that cabled knot detection and composite knot detection are unconditionally in NP. Our algorithms are based on recent work of Kuperberg and of Lackenby on detecting knottedness.
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Additional Information
  • John A. Baldwin
  • Affiliation: Department of Mathematics, Boston College, Maloney Hall, Fifth Floor, Boston College, Chestnut Hill, Massachusetts 02467-3806
  • MR Author ID: 772542
  • Email:
  • Steven Sivek
  • Affiliation: Department of Mathematics, Imperial College London, Huxley Building, 180 Queen’s Gate, London SW7 2AZ, United Kingdom
  • MR Author ID: 781378
  • Email:
  • Received by editor(s): June 16, 2017
  • Received by editor(s) in revised form: September 6, 2017, and September 9, 2017
  • Published electronically: November 16, 2018
  • Additional Notes: The first author was supported by NSF Grant DMS-1406383 and NSF CAREER Grant DMS-1454865.
    The second author was supported by the Max Planck Institute for Mathematics during some of the period in which this paper was completed.
  • © Copyright 2018 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 371 (2019), 3831-3855
  • MSC (2010): Primary 57M25; Secondary 68Q17
  • DOI:
  • MathSciNet review: 3917210