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Detecting Unknots via Equational Reasoning, I: Exploration

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

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.

Keywords

  • Fundamental Group
  • Homomorphic Image
  • Automate Reasoning
  • Automate Theorem Prove
  • Automate Deduction

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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