Multi-virtual braid groups

TL;DR

This paper introduces new classes of multi-virtual braid groups, explores their algebraic structures, and establishes their relations to existing groups, advancing the understanding of braid group generalizations.

Contribution

It defines multi-virtual pure and semi-pure braid groups, provides generators and relations, and analyzes their structure as semi-direct products, extending the theory of virtual braid groups.

Findings

Multi-virtual pure and semi-pure braid groups are normal subgroups of index n!

Multi-virtual braid groups are semi-direct products of pure groups and symmetric groups

Structural analysis of three-strand 2-virtual braid group and its subgroups

Abstract

L. Kauffman (2024) introduced multi-virtual and symmetric multi-virtual braid groups, which are generalizations of the virtual braid group. We introduce multi-virtual pure and multi-virtual semi-pure braid groups, which are normal subgroups of index $n!$. We give a set of generators and defining relations for these groups, show that multi-virtual (symmetric multi-virtual) braid group is a semi-direct products of multi-virtual pure (symmetric multi-virtual pure) braid group and symmetric group. Also, we introduce multi-welded and multi-unrestricted braid groups and examines structure of three-strand 2-virtual braid group and some its subgroups and quotients. The paper concludes by outlining open problems and suggesting avenues for future research in this area.