Dual Garside structure and reducibility of braids

TL;DR

This paper provides a new geometric proof relating the properties of braids as homeomorphisms to their algebraic Garside structures, and introduces an algorithm for classifying braid types.

Contribution

It offers a simplified geometric proof of a key result and extends it to the dual Garside structure, enabling a new classification algorithm for braids.

Findings

New geometric proof of the Benardete-Gutierrez-Nitecki result

Extension of the result to dual Garside structure

Development of a new braid classification algorithm

Abstract

Benardete, Gutierrez and Nitecki showed an important result which relates the geometrical properties of a braid, as a homeomorphism of the punctured disk, to its algebraic Garside-theoretical properties. Namely, they showed that if a braid sends a curve to another curve, then the image of this curve after each factor of the left normal form of the braid (with the classical Garside structure) is also standard. We provide a new simple, geometric proof of the result by Benardete-Gutierrez-Nitecki, which can be easily adapted to the case of the dual Garside structure of braid groups, with the appropriate definition of standard curves in the dual setting. This yields a new algorithm for determining the Nielsen-Thurston type of braids.