Twisted Virtual Braid Group

TL;DR

This paper explores the structure of the twisted virtual braid group, identifying key subgroups, their generators, relations, and decompositions, including semi-direct products and connections to virtual pure braid groups.

Contribution

It provides explicit generators, relations, and decompositions for the twisted virtual pure braid group and related subgroups, extending the understanding of their algebraic structure.

Findings

$TVP_n$ is a semi-direct product of a subgroup and $bZ_2^n$

$TVB_n$ decomposes as $TVP_n times S_n$

Constructed new epimorphisms and identified kernels with explicit presentations

Abstract

In this paper we study some subgroups and their decompositions in semi-direct product of the twisted virtual braid group $TVB_n$. In particular, the twisted virtual pure braid group $TVP_n$ is the kernel of an epimorphism of $TVB_n$ onto the symmetric group $S_n$. We find the set of generators and defining relations for $TVP_n$ and show that $TVB_n = TVP_n \rtimes S_n$. Further we prove that $TVP_n$ is a semi-direct product of some subgroup and abelian group $\mathbb{Z}_2^n$. As corollary we get that the virtual pure braid group $VP_n$ is a subgroup of $TVP_n$. Also, we construct some other epimorphism of $TVB_n$ onto $S_n$. Its kernel, $TVH_n$ is an analogous of $TVP_n$. We find its set of generators and defining relations and construct its decomposition in a semi-direct product.