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Describing semigroups with defining relations of the form xy=yz xy and yx=zy and connections with knot theory

Vernitski, Alexei (2017) 'Describing semigroups with defining relations of the form xy=yz xy and yx=zy and connections with knot theory.' Semigroup Forum, 95 (1). 66 - 82. ISSN 0037-1912

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Abstract

We introduce a knot semigroup as a cancellative semigroup whose defining relations are produced from crossings on a knot diagram in a way similar to the Wirtinger presentation of the knot group; to be more precise, a knot semigroup as we define it is closely related to such tools of knot theory as the twofold branched cyclic cover space of a knot and the involutory quandle of a knot. We describe knot semigroups of several standard classes of knot diagrams, including torus knots and torus links T(2, n) and twist knots. The description includes a solution of the word problem. To produce this description, we introduce alternating sum semigroups as certain naturally defined factor semigroups of free semigroups over cyclic groups. We formulate several conjectures for future research.

Item Type: Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science and Health > Mathematical Sciences, Department of
Depositing User: Jim Jamieson
Date Deposited: 19 May 2016 12:58
Last Modified: 20 Jul 2018 16:15
URI: http://repository.essex.ac.uk/id/eprint/16729

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