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2017 | OriginalPaper | Buchkapitel

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

verfasst von : Alexei Lisitsa, Alexei Vernitski

Erschienen in: Mathematical Aspects of Computer and Information Sciences

Verlag: Springer International Publishing

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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.

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Nächstes Kapitel Balancing Expression Dags for More Efficient Lazy Adaptive Evaluation

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.

Fatal Error: mace4: domain_element too big.

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

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.

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.

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

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Titel: Automated Reasoning for Knot Semigroups and -orbifold Groups of Knots
verfasst von: Alexei Lisitsa
Alexei Vernitski
Verlag: Springer International Publishing
Buch: Mathematical Aspects of Computer and Information Sciences
Print ISBN: 978-3-319-72452-2

Electronic ISBN: 978-3-319-72453-9

Copyright-Jahr: 2017
DOI: https://doi.org/10.1007/978-3-319-72453-9_1

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