On some torus knot groups and submonoids of the braid groups

Abstract

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)n≥1 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,…,σ1σ2⋯σn, 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.