A journey through loop braid groups

TL;DR

This paper unifies various definitions of loop braid groups, a generalization of braid groups with applications in mathematics and physics, and introduces a natural extension of these groups.

Contribution

It provides a comprehensive unification of all existing definitions of loop braid groups and introduces a new, topologically motivated extension.

Findings

Proved the equivalence of different definitions of loop braid groups.

Unified terminology and concepts across multiple mathematical domains.

Introduced a natural extension of loop braid groups.

Abstract

In this paper we introduce distinct approaches to loop braid groups, a generalisation of braid groups, and unify all the definitions that have appeared so far in literature, with a complete proof of the equivalence of these definitions. These groups have in fact been an object of interest in different domains of mathematics and mathematical physics, and have been called, in addition to loop braid groups, with several names such as of motion groups, groups of permutation-conjugacy automorphisms, braid-permutation groups, welded braid groups and untwisted ring groups. In parallel to this, we introduce an extension of these groups that appears to be a more natural generalisation of braid groups from the topological point of view. Throughout the text we motivate the interest in studying loop braid groups and give references to some of their applications.