SIMPLE DUAL BRAIDS, NONCROSSING PARTITIONS AND MIKADO BRAIDS OF TYPE Dn

We show that the simple elements of the dual Garside structure of an Artin group of type Dn are Mikado braids, giving a positive answer to a conjecture of Digne and the second author. To this end, we use an embedding of the Artin group of type Dn in a suitable quotient of an Artin group of type Bn noticed by Allcock, of which we give a simple algebraic proof here. This allows one to give a characterization of the Mikado braids of type Dn in terms of those of type Bn and also to describe them topologically. Using this topological representation and Athanasiadis and Reiner’s model for noncrossing partitions of type Dn which can be used to represent the simple elements, we deduce the above mentioned conjecture. AMS 2010 Mathematics Classification: : 20F36, 20F55.