The monoid structure of singular twisted virtual braids

TL;DR

This paper explores the algebraic structure of the singular twisted virtual braid monoid, identifying key submonoids, their relations, and embeddings, and extends braid group representations to this monoid.

Contribution

It introduces the singular twisted virtual pure braid monoid as a kernel, describes its generators and relations, and extends braid group representations to this monoid.

Findings

$STVP_n$ is the kernel of an epimorphism from $STVB_n$ to $S_n$

The monoid $STVB_n$ embeds into a group

Representation of the braid group extends to the monoid

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.