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Algebraic structures among virtual singular braids

Carmen Caprau, Antonia Yeung

Abstract

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

Algebraic structures among virtual singular braids

Abstract

We show that the virtual singular braid monoid on strands embeds in a group , which we call the virtual singular braid group on strands. The group contains a normal subgroup of virtual singular pure braids. We show that is a semi-direct product of and the symmetric group . We provide a presentation for via generators and relations. We also represent as a semi-direct product of subgroups and study the structures of these subgroups. These results yield a normal form of words in the virtual singular braid group.
Paper Structure (5 sections, 15 theorems, 62 equations, 2 figures)

This paper contains 5 sections, 15 theorems, 62 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Virtual singular braid monoid embeds in a group
  3. Another presentation for the virtual singular braid group
  4. Generators and relations of the virtual singular pure braid group
  5. The virtual singular pure braid group as a semi direct product

Key Result

Theorem 2

Keyman Let ${\mathcal{M}}$ be a monoid given by a presentation $[A \cup B \, | \,{\bf R}]$ where $A=\{ a_1, \dots, a_n \}$, $B=\{ b_1, \dots, b_m \}$ and ${\bf R = R_1 \cup R_2 \cup R_3 \cup R_4}$, where: Then ${\mathcal{M}}$ embeds in a group ${\mathcal{G}}$ with presentation $[A \cup B \cup \bar{B} \, | \, {\bf R \cup R' }]$, where $\bar{B}=\{ \bar{b}_j | j = 1, \dots, m \}$ and ${\bf R'}$ c

Figures (2)

  • Figure 1: The braids $\mu_{i, i+1}, \, \mu_{i, i+1}^{-1}, \gamma_{i, i+1}$ and $\bar{\gamma}_{i, i+1}$
  • Figure 2: Braids representing $\mu_{ij}, \, \mu_{ji}, \, \gamma_{ij}$ and $\gamma_{ji}$, for $i < j$

Theorems & Definitions (34)

  • Definition 1
  • Theorem 2
  • Definition 3
  • Theorem 4
  • proof
  • Definition 5
  • Definition 6
  • Definition 7
  • Lemma 8
  • proof
  • ...and 24 more