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

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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.

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Notes

  1. Note that a knot semigroup is not a knot invariant; that is, there are cases when two different diagrams of the same knot produce non-isomorphic knot semigroups. See more in Sect. 8.

  2. The property of being thin relative to knot semigroups can be reformulated as follows: if two words \(u=au'\) and \(v=bv'\) are equal, where a, b are letters, then there are letters c, d such that \(ac=bd\) and \(u=acu''\) and \(v=bdv''\).

  3. It is known, see, for example, [15], that a presentation of a cancellative semigroup can be considerably more complicated than its cancellative presentation.

  4. Reidemeister moves are standard ways of transforming a knot diagram to produce other diagrams of the same knot.

  5. Note that the term ‘braid semigroup’ already has a well established meaning, namely, it is the subsemigroup of the braid group generated by clockwise half-twists (see, for example, [2]). To avoid confusion, we do not use the words ‘braid semigroup’ to mean the knot semigroup of a braid.

  6. A meridian is an element of the knot group corresponding to one arc.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Essex, Colchester, UK

    Alexei Vernitski

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  1. Alexei Vernitski
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Correspondence to Alexei Vernitski.

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Communicated by Victoria Gould.

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Vernitski, A. Describing semigroups with defining relations of the form \(xy=yz\) and \(yx=zy\) and connections with knot theory. Semigroup Forum 95, 66–82 (2017). https://doi.org/10.1007/s00233-016-9808-7

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