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International Conference on Mathematical Aspects of Computer and Information Sciences

MACIS 2017: Mathematical Aspects of Computer and Information Sciences pp 3–18Cite as

Automated Reasoning for Knot Semigroups and \(\pi \)-orbifold Groups of Knots

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10693))

Abstract

The paper continues the first author’s research which shows that automatic reasoning is an effective tool for establishing properties of algebraic constructions associated with knot diagrams. Previous research considered involutory quandles (also known as keis) and quandles. This paper applies automated reasoning to knot semigroups, recently introduced and studied by the second author, and \(\pi \)-orbifold groups of knots. We test two conjectures concerning knot semigroups (specifically, conjectures aiming to describe knot semigroups of diagrams of the trivial knot and knot semigroups of 4-plat knot diagrams) on a large number of examples. These experiments enable us to formulate one new conjecture. We discuss applications of our results to a classical problem of the knot theory, determining whether a knot diagram represents the trivial knot.

Alexei Lisitsa—Part of this research was carried out during visits by the first named author to the Department of Mathematical Sciences at the University of Essex in 2016. The visits were financed by a London Mathematical Society Scheme 7 Grant (ref. SC7-1516-12). This research has been supported by the European Union’s Horizon 2020 research and innovation programme under grant agreement No. H2020-FETOPEN-2015-CSA 712689 (SC \(^2\)).

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Notes

  1. 1.

    We have chosen Prover9 and model builder Mace4 (below), primarily to be able to compare efficiency of automated reasoning with semigroups with that for involutory quandles in [6], where the same systems were used. Otherwise the choice is not very essential and any other automated first order (dis)provers could be used instead.

  2. 2.

    Fatal Error: mace4: domain_element too big.

  3. 3.

    The described procedure is a version of so-called Fox coloring [17]. Note that in general, labels of some distinct arcs may coincide.

  4. 4.

    We are grateful to José Montesinos (Universidad Complutense de Madrid), Genevieve Walsh (Tufts University) and Vanni Noferini (University of Essex) for attracting our attention to this result.

  5. 5.

    Note that here we mean the usual semigroup deduction, not a more complicated one used in cancellative semigroups. It is useful to remind oneself of this, because knot semigroups are defined using a cancellative presentation, and it makes proving equalities of words in knot semigroups more involved.

  6. 6.

    https://zenodo.org/record/1009577, https://doi.org/10.5281/zenodo.1009577.

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Author information

Authors and Affiliations

  1. Department of Computer Science, University of Liverpool, Liverpool, UK

    Alexei Lisitsa

  2. Department of Mathematical Sciences, University of Essex, Essex, UK

    Alexei Vernitski

Authors
  1. Alexei Lisitsa
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  2. Alexei Vernitski
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Corresponding authors

Correspondence to Alexei Lisitsa or Alexei Vernitski .

Editor information

Editors and Affiliations

  1. University of Paderborn, Paderborn, Germany

    Johannes Blömer

  2. Wilfrid Laurier University, Waterloo, Ontario, Canada

    Ilias S. Kotsireas

  3. Johannes Kepler University Linz, Linz, Austria

    Temur Kutsia

  4. SBA Research, Vienna, Austria

    Dimitris E. Simos

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Lisitsa, A., Vernitski, A. (2017). Automated Reasoning for Knot Semigroups and \(\pi \)-orbifold Groups of Knots. In: Blömer, J., Kotsireas, I., Kutsia, T., Simos, D. (eds) Mathematical Aspects of Computer and Information Sciences. MACIS 2017. Lecture Notes in Computer Science(), vol 10693. Springer, Cham. https://doi.org/10.1007/978-3-319-72453-9_1

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  • DOI: https://doi.org/10.1007/978-3-319-72453-9_1

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  • Online ISBN: 978-3-319-72453-9

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