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

Describing semigroups with defining relations of the form $$xy=yz$$ x y = y z and... 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Semigroup Forum Springer Journals

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

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Publisher
Springer US
Copyright
Copyright © 2016 by Springer Science+Business Media New York
Subject
Mathematics; Algebra
ISSN
0037-1912
eISSN
1432-2137
D.O.I.
10.1007/s00233-016-9808-7
Publisher site
See Article on Publisher Site

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.

Journal

Semigroup ForumSpringer Journals

Published: Jun 9, 2016

References

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