A new Garside structure on torus knot groups and some complex braid groups

TL;DR

This paper introduces a new Garside monoid structure for all torus knot groups, extending previous work, and also constructs new Garside structures for certain complex braid groups of rank two.

Contribution

It generalizes Garside structures to all torus knot groups and provides new structures for specific complex reflection group braid groups.

Findings

New Garside monoid for all (n,m)-torus knot groups

Garside structures for exceptional complex reflection group braid groups

Extensions to non-torus complex reflection groups G_{13} and dihedral Artin groups

Abstract

Several distinct Garside monoids having torus knot groups as groups of fractions are known. For $n,m\geq 2$ two coprime integers, we introduce a new Garside monoid $\mathcal{M}(n,m)$ having as Garside group the $(n,m)$-torus knot group, thereby generalizing to all torus knot groups a construction that we previously gave for the $(n,n+1)$-torus knot group. As a byproduct, we obtain new Garside structures for the braid groups of a few exceptional complex reflection groups of rank two. Analogous Garside structures are also constructed for a few additional braid groups of exceptional complex reflection groups of rank two which are not isomorphic to torus knot groups, namely for $G_{13}$ and for dihedral Artin groups of even type.