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Multi-virtual braid groups

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

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.

Multi-virtual braid groups

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 . 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.

Paper Structure

This paper contains 12 sections, 11 theorems, 110 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Basic definitions
  3. Braid group
  4. Virtual braid group
  5. Knot theory and multi-virtual knot theory
  6. Multi-virtual braid group
  7. Multi-virtual pure braid group
  8. Multi virtual semi--pure braid group
  9. Symmetric multi-virtual braid group
  10. Symmetric multi-virtual pure braid group
  11. Symmetric multi-virtual semi-pure braid group
  12. Some quotients and subgroups of multi-virtual braid group

Key Result

Lemma 2.1

Let $a$ be an element of $\langle \rho_1, \rho_2, \ldots, \rho_{n-1} \rangle$ and $\bar{a}$ is its image in $S_n$ under the isomorphism $\rho_i \mapsto (i,i+1)$, $i = 1, 2, \ldots, n-1$, then for any generator $\lambda_{ij}$ of $VP_n$ the following holds where $(k)\bar{a}$ is the image of $k$ under the action of the permutation $\bar{a}$.

Figures (7)

  • Figure 1: The geometric interpretation of $\sigma_i$ and $\sigma_i^{-1}$
  • Figure 2: Geometric interpretations of generators $\lambda_{ij}$ and $\lambda_{ji}$
  • Figure 3: Reidemeister and virtual Reidemeister moves
  • Figure 4: Detour move for a virtual crossings of type $0$
  • Figure 5: Forbidden detour move for a virtual crossings of type $\alpha \not= 0$
  • ...and 2 more figures

Theorems & Definitions (21)

  • Lemma 2.1: B
  • Example 2.2
  • Lemma 2.3: BB
  • Remark 3.1
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Lemma 3.4
  • Theorem 3.5
  • proof
  • ...and 11 more