Twisted Virtual Braid Group

Valeriy G. Bardakov; Tatyana A. Kozlovskaya; Komal Negi; Madeti Prabhakar

paper

Twisted Virtual Braid Group

arXiv:2310.04154

Abstract

In this paper we study some subgroups and their decompositions in semi-direct product of the twisted virtual braid group $T V B_{n}$ . In particular, the twisted virtual pure braid group $T V P_{n}$ is the kernel of an epimorphism of $T V B_{n}$ onto the symmetric group $S_{n}$ . We find the set of generators and defining relations for $T V P_{n}$ and show that $T V B_{n} = T V P_{n} ⋊ S_{n}$ . Further we prove that $T V P_{n}$ is a semi-direct product of some subgroup and abelian group $Z_{2}^{n}$ . As corollary we get that the virtual pure braid group $V P_{n}$ is a subgroup of $T V P_{n}$ . Also, we construct some other epimorphism of $T V B_{n}$ onto $S_{n}$ . Its kernel, $T V H_{n}$ is an analogous of $T V P_{n}$ . We find its set of generators and defining relations and construct its decomposition in a semi-direct product.

20 pages, 2 figures, uses overleaf