Algebraic Structures Among Virtual Singular Braids

期刊论文
发表日期：

2024年6月25日

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Carmen Livia Caprau,
Antonia Yeung

摘要

We show that the virtual singular braid monoid on n strands embeds in a group $VSG_n$ $V S G_{n}$ , which we call the virtual singular braid group on n strands. The group $VSG_n$ $V S G_{n}$ contains a normal subgroup $VSPG_n$ $V S P G_{n}$ of virtual singular pure braids. We show that $VSG_n$ $V S G_{n}$ is a semi-direct product of $VSPG_n$ $V S P G_{n}$ and the symmetric group $S_n$ $S_{n}$ . We provide a presentation for $VSPG_n$ $V S P G_{n}$ via generators and relations. We also represent $VSPG_n$ $V S P G_{n}$ as a semi-direct product of $n-1$ $n - 1$ subgroups and study the structures of these subgroups. These results yield a normal form of words in the virtual singular braid group.