Reduce Problems From Braid Groups To Braid Monoids

TL;DR

This paper introduces linear time reductions of the word and conjugacy problems in braid groups to their monoids, simplifying problem-solving by focusing on positive words representations.

Contribution

It provides the first linear time reductions from braid groups to braid monoids for key computational problems, using only positive words.

Findings

Reductions operate in linear time.

Problems are reduced to positive words representations.

Simplifies computational approaches to braid groups.

Abstract

This paper proposes for every $n$, linear time reductions of the word and conjugacy problems on the braid groups $B_n$ to the corresponding problems on the braid monoids $B_n^+$ and moreover only using positive words representations.