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Detecting Unknots via Equational Reasoning, I: Exploration

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Intelligent Computer Mathematics (CICM 2014)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 8543))

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Abstract

We explore the application of automated reasoning techniques to unknot detection, a classical problem of computational topology. We adopt a two-pronged experimental approach, using a theorem prover to try to establish a positive result (i.e. that a knot is the unknot), whilst simultaneously using a model finder to try to establish a negative result (i.e. that the knot is not the unknot). The theorem proving approach utilises equational reasoning, whilst the model finder searches for a minimal size counter-model. We present and compare experimental data using the involutary quandle of the knot, as well as comparing with alternative approaches, highlighting instances of interest. Furthermore, we present theoretical connections of the minimal countermodels obtained with existing knot invariants, for all prime knots of up to 10 crossings: this may be useful for developing advanced search strategies.

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Author information

Authors and Affiliations

  1. School of Computing, Engineering and Mathematics, University of Brighton, UK

    Andrew Fish

  2. Department of Computer Science, The University of Liverpool, UK

    Alexei Lisitsa

Authors
  1. Andrew Fish
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  2. Alexei Lisitsa
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Editors and Affiliations

  1. Computer Science Department, The University of Western Ontario, MC 375,, N6A 5B7, London, ON, Canada

    Stephen M. Watt

  2. Departments of Computer Science and Mathematical Sciences, University of Bath, Claverton Down, BA2 7AY, Bath, UK

    James H. Davenport

  3. School of Computer Science, Edgbaston, University of Birmingham, B15 2TT, Birmingham, UK

    Alan P. Sexton

  4. Faculty of Informatics, Masaryk University, Botanicá 6a, 60200, Brno, Czech Republic

    Petr Sojka

  5. Institute for Computing Sciences, Radboud University Nijmegen, Postbus 9010,, 6525, Nijmegen, GL, The Netherlands

    Josef Urban

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Fish, A., Lisitsa, A. (2014). Detecting Unknots via Equational Reasoning, I: Exploration. In: Watt, S.M., Davenport, J.H., Sexton, A.P., Sojka, P., Urban, J. (eds) Intelligent Computer Mathematics. CICM 2014. Lecture Notes in Computer Science(), vol 8543. Springer, Cham. https://doi.org/10.1007/978-3-319-08434-3_7

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  • DOI: https://doi.org/10.1007/978-3-319-08434-3_7

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-08433-6

  • Online ISBN: 978-3-319-08434-3

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