On torus knot groups and a submonoid of the braid group - Archive ouverte HAL
Pré-Publication, Document De Travail Année : 2022

On torus knot groups and a submonoid of the braid group

Thomas Gobet
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Résumé

The submonoid of the 3-strand braid group B3 generated by σ1 and σ1σ2 is known to yield an exotic Garside structure on B3. We introduce and study an infinite family (Mn)n1 of Garside monoids generalizing this exotic Garside structure, i.e., such that M2 is isomorphic to the above monoid. The corresponding Garside group G(Mn) is isomorphic to the (n,n+1)-torus knot group-which is isomorphic to B3 for n=2 and to the braid group of the exceptional complex reflection group G12 for n=3. This yields a new Garside structure on (n,n+1)-torus knot groups, which already admit several distinct Garside structures. The (n,n+1)-torus knot group is an extension of Bn+1, and the Garside monoid Mn surjects onto the submonoid Σn of Bn+1 generated by σ1,σ1σ2,,σ1σ2σn, which is not a Garside monoid when n>2. Using a new presentation of Bn+1 that is similar to the presentation of G(Mn), we nevertheless check that Σn is an Ore monoid with group of fractions isomorphic to Bn+1, and give a conjectural presentation of it, similar to the defining presentation of Mn. This partially answers a question of Dehornoy-Digne-Godelle-Krammer-Michel.
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hal-02940881 , version 1 (22-07-2024)

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Thomas Gobet. On torus knot groups and a submonoid of the braid group. 2020. ⟨hal-02940881⟩
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