Abstract
It is known that a knot (link) is rational if and only if its \(\pi \)-orbifold group is dihedral. A semigroup version of this result has been formulated as a conjecture. Working towards proving the conjecture, we describe certain semigroups associated with twist links, clarify how these semigroups are related to dihedral groups and find defining relations of these semigroups.
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Notes
Some minor variations of the constructions are possible, which do not affect the knot semigroups. For instance, the torus knot (link) can be considered as one represented by a 4-plat diagram whose braid part is \(\sigma _{1}^{-n}\) instead of \(\sigma _{1}^{n}\), as discussed in the beginning of Sect. 4 in [10].
We are grateful to Jose Montesinos (Universidad Complutense de Madrid), Genevieve Walsh (Tufts University) and Vanni Noferini (University of Essex) for attracting our attention to this result.
For understanding the structure of the proof, it is useful to note that this is the only place where we use relations at both ends of the twist diagram to produce an effect of treating indices as numbers modulo \(4(c+d)+4\).
More generally, links \(\mathfrak {tw}_{c}^{d}\) and \(\mathfrak {tw}_{c+d}^{0}\) are equivalent for all values of c and d.
For example, the main theorem in [2] (Sect. 5.2) showing that quandles are knot invariants excludes split links from consideration.
Two of the authors (A. Vernitski and A. Lisitsa) are now involved in research in which the process of untangling a knot diagram of the trivial knot is facilitated by considering knot semigroups of certain tangle diagrams.
Chapter 5 of [9] contains a useful discussion of how proofs in cancellative semigroups compare with those in groups.
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Communicated by Victoria Gould.
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Vernitski, A., Tunsi, L., Ponchel, C. et al. Dihedral semigroups, their defining relations and an application to describing knot semigroups of rational links. Semigroup Forum 97, 75–86 (2018). https://doi.org/10.1007/s00233-018-9918-5
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DOI: https://doi.org/10.1007/s00233-018-9918-5