About presentations of braid groups and their generalizations

TL;DR

This paper surveys recent generalizations of braid groups, including inverse monoids and alternative presentations, highlighting new structures and results in the theory of braids.

Contribution

It introduces various new notions and presentations of braid groups, including inverse monoids, Sergiescu graph-presentations, and Birman-Ko-Lee presentations, expanding the classical theory.

Findings

Different presentations of braid groups with fewer generators

Analysis of inverse monoids of partial braids

Results on word and conjugacy problems for singular braid monoids

Abstract

In the paper we give a survey of rather new notions and results which generalize classical ones in the theory of braids. Among such notions are various inverse monoids of partial braids. We also observe presentations different from standard Artin presentation for generalizations of braids. Namely, we consider presentations with small number of generators, Sergiescu graph-presentations and Birman-Ko-Lee presentation. The work of V.V.Chaynikov on the word and conjugacy problems for the singular braid monoid in Birman-Ko-Lee generators is described as well.