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The monoid structure of singular twisted virtual braids

Prabhakar Madeti, Komal Negi

TL;DR

This work analyzes the algebraic structure of the singular twisted virtual braid monoid $STVB_n$ by constructing and classifying important submonoids via epimorphisms to the symmetric group $S_n$, including kernels STVP$_n$, STVH$_n$, and $M_n$, each with explicit presentations. It proves that STVB$_n$ embeds into a group $STVG_n$ with a pure subgroup $STVPG_n$, and derives the corresponding group presentations; an epimorphism to $S_n$ is used to characterize these submonoids. The paper also extends existing representations from the twisted virtual braid group $TVB_n$ to the singular setting, providing a concrete method to lift representations to $STVB_n$ via linear combinations for the singular generators. Collectively, these results advance the understanding of the algebraic and representation-theoretic structure of singular twisted virtual braids and set the stage for further decomposition and isomorphism questions among their submonoids. The findings have potential implications for the study of singular twisted virtual links and their associated algebraic invariants.

Abstract

In this paper, we examine specific submonoids within the singular twisted virtual braid monoid $STVB_n$. Notably, we establish that the singular twisted virtual pure braid monoid $STVP_n$ serves as the kernel of an epimorphism from $STVB_n$ onto the symmetric group $S_n$. We identify the generators and defining relations for $STVP_n$. Additionally, we construct other epimorphisms from $STVB_n$ onto $S_n$, whose kernels are analogous to $STVP_n$, and determine their respective generators and defining relations. Furthermore, we demonstrate the embedding of the monoid $STVB_n$ into a group. Also, we provide the extension of the representation of the twisted virtual braid group to the representation of the singular twisted virtual braid monoid.

The monoid structure of singular twisted virtual braids

TL;DR

This work analyzes the algebraic structure of the singular twisted virtual braid monoid by constructing and classifying important submonoids via epimorphisms to the symmetric group , including kernels STVP, STVH, and , each with explicit presentations. It proves that STVB embeds into a group with a pure subgroup , and derives the corresponding group presentations; an epimorphism to is used to characterize these submonoids. The paper also extends existing representations from the twisted virtual braid group to the singular setting, providing a concrete method to lift representations to via linear combinations for the singular generators. Collectively, these results advance the understanding of the algebraic and representation-theoretic structure of singular twisted virtual braids and set the stage for further decomposition and isomorphism questions among their submonoids. The findings have potential implications for the study of singular twisted virtual links and their associated algebraic invariants.

Abstract

In this paper, we examine specific submonoids within the singular twisted virtual braid monoid . Notably, we establish that the singular twisted virtual pure braid monoid serves as the kernel of an epimorphism from onto the symmetric group . We identify the generators and defining relations for . Additionally, we construct other epimorphisms from onto , whose kernels are analogous to , and determine their respective generators and defining relations. Furthermore, we demonstrate the embedding of the monoid into a group. Also, we provide the extension of the representation of the twisted virtual braid group to the representation of the singular twisted virtual braid monoid.

Paper Structure

This paper contains 11 sections, 10 theorems, 40 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Singular twisted virtual braid monoid
  4. Some submonoids of $STVB_n$
  5. Structure of singular twisted virtual pure braid monoid $STVP_n$.
  6. The submonoid $STVH_n$ of monoid $STVB_n$
  7. The submonoid $M_n$ of monoid $STVB_n$
  8. Some more results
  9. Embedding of monoid $STVB_n$ in a group $STVG_n$
  10. Extension of representation of $TVB_n$ to $STVB_n$
  11. Conclusion

Key Result

Theorem 2.4

KM The monoid $STVB_n$ is generated by standard generators, $\sigma_i^{\pm 1}$, $\rho_i$, $\tau_i$$(i=1, \dots, n-1)$, and $\gamma_i$$(i=1, \dots, n)$ satisfy the following relations, where $e$ represents the identity element:

Figures (8)

  • Figure 1: A singular twisted virtual braid on 3 strands
  • Figure 2: Classical braid moves
  • Figure 3: Virtual braid moves
  • Figure 4: Twisted braid moves
  • Figure 5: Singular braid moves
  • ...and 3 more figures

Theorems & Definitions (18)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Remark 2.5
  • Theorem 2.6
  • Lemma 3.1
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • ...and 8 more