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

2022/09/06 by Gobet, Thomas
#FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2209.02291

Abstract

Several distinct Garside monoids having torus knot groups as groups of fractions are known. For n,m≥ 2 two coprime integers, we introduce a new Garside monoid 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 G₁₃ and for dihedral Artin groups of even type.

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• Faithful Burau-like representations of some rank two Garside groups and torus knot groups

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