# Simple dual braids, noncrossing partitions and Mikado braids of type $D_n$

## TL;DR

This paper proves that simple elements in the dual Garside structure of type D_n Artin groups are Mikado braids, confirming a conjecture by Digne and Reiner through algebraic and topological methods.

## Contribution

It establishes that simple elements are Mikado braids in type D_n, linking noncrossing partitions, Mikado braids, and dual Garside structures, and provides an algebraic proof of an embedding.

## Key findings

- Simple elements of type D_n are Mikado braids.
- An algebraic proof of the embedding of type D_n into type B_n.
- Topological characterization of Mikado braids in type D_n.

## Abstract

We show that the simple elements of the dual Garside structure of an Artin group of type $D_n$ 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 $D_n$ in a suitable quotient of an Artin group of type $B_n$ 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 $D_n$ in terms of those of type $B_n$ and also to describe them topologically. Using this topological representation and Athanasiadis and Reiner's model for noncrossing partitions of type $D_n$ which can be used to represent the simple elements, we deduce the above mentioned conjecture.