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Learning to unknot

Gukov, Sergei and Halverson, James and Ruehle, Fabian and Sułkowski, Piotr (2021) Learning to unknot. Machine Learning: Science and Technology, 2 (2). Art. No. 025035. ISSN 2632-2153. doi:10.1088/2632-2153/abe91f.

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We introduce natural language processing into the study of knot theory, as made natural by the braid word representation of knots. We study the UNKNOT problem of determining whether or not a given knot is the unknot. After describing an algorithm to randomly generate N-crossing braids and their knot closures and discussing the induced prior on the distribution of knots, we apply binary classification to the UNKNOT decision problem. We find that the Reformer and shared-QK Transformer network architectures outperform fully-connected networks, though all perform at ≳95% accuracy. Perhaps surprisingly, we find that accuracy increases with the length of the braid word, and that the networks learn a direct correlation between the confidence of their predictions and the degree of the Jones polynomial. Finally, we utilize reinforcement learning (RL) to find sequences of Markov moves and braid relations that simplify knots and can identify unknots by explicitly giving the sequence of unknotting actions. Trust region policy optimization (TRPO) performs consistently well, reducing ≳80% of the unknots with up to 96 crossings we tested to the empty braid word, and thoroughly outperformed other RL algorithms and random walkers. Studying these actions, we find that braid relations are more useful in simplifying to the unknot than one of the Markov moves.

Item Type:Article
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URLURL TypeDescription Paper
Gukov, Sergei0000-0002-9486-1762
Halverson, James0000-0003-0535-2622
Ruehle, Fabian0000-0002-8409-9823
Sułkowski, Piotr0000-0002-6176-6240
Additional Information:© 2021 The Author(s). Published by IOP Publishing Ltd. Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 license. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Received 9 November 2020; Accepted 23 February 2021; Published 21 April 2021. We thank Peter Battaglia, Kyle Cranmer, Michael Freedman, Mark Hughes, Ciprian Manolescu, Alex Radovic, Danilo Rezende, and Adam Sikora for useful discussions. The work of S.G. is supported by the U.S. Department of Energy, Office of Science, Office of High Energy Physics, under Award No. DE-SC0011632, and by the National Science Foundation under Grant No. NSF DMS 1664227. J.H. is supported by NSF CAREER grant PHY-1848089. The work of P.S. is supported by the TEAM programme of the Foundation for Polish Science co-financed by the European Union under the European Regional Development Fund (POIR.04.04.00-00-5C55/17-00). Data availability statement: The data that support the findings of this study are available upon reasonable request from the authors.
Group:Walter Burke Institute for Theoretical Physics
Funding AgencyGrant Number
Department of Energy (DOE)DE-SC0011632
Foundation for Polish ScienceUNSPECIFIED
European Regional Development FundPOIR.04.04.00-00-5C55/17-00
Subject Keywords:knot theory, string theory, machine learning, reinforcement learning
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Other Numbering System NameOther Numbering System ID
Issue or Number:2
Record Number:CaltechAUTHORS:20201111-131628016
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Official Citation:Sergei Gukov et al 2021 Mach. Learn.: Sci. Technol. 2 025035
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:106623
Deposited By: Joy Painter
Deposited On:11 Nov 2020 23:24
Last Modified:28 Jun 2021 21:25

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