On the centralizer of generic braids

TL;DR

This paper investigates the structure of braid centralizers using Garside theory, introducing an efficient algorithm that computes minimal generators with quadratic complexity in typical cases.

Contribution

It presents a novel algorithm for computing braid centralizers efficiently, leveraging the structure of ultra summit sets in generic braids.

Findings

Minimal generating sets can be computed efficiently for generic braids.

The ultra summit set of a generic braid has a specific structure that facilitates this computation.

The proposed algorithm has quadratic complexity in the length of the braid in the generic case.

Abstract

We study the centralizer of a braid from the point of view of Garside theory, showing that generically a minimal set of generators can be computed very efficiently, as the ultra summit set of a generic braid has a very particular structure. We present an algorithm to compute the centralizer of a braid whose generic-case complexity is quadratic on the length of the input, and which outputs a minimal set of generators in the generic case.