# Algebraic, combinatorial and topological properties of singular virtual braid monoids

## TL;DR

This paper explores the algebraic, combinatorial, and topological aspects of singular virtual braid monoids, establishing new relations, conjectures, and bijections with diagrams and presentations.

## Contribution

It introduces a Birman-like conjecture for virtual singular braids and establishes a bijection between singular abstract braids, Gauss diagrams, and virtual braids.

## Key findings

- Bijection between singular abstract braids and Gauss diagrams
- Presentation of the singular pure virtual braid monoid
- Discussion of a Birman-like conjecture for virtual singular braids

## Abstract

In this paper we discuss algebraic, combinatorial and topological properties of singular virtual braids. On the algebraic side we state the relations between classical and virtual singular objects, in addition we discuss a Birman-like conjecture for the virtual case. On the topological and combinatorial side, we prove that there is a bijection between singular abstract braids, horizontal Gauss diagrams and singular virtual braids, in particular using horizontal Gauss diagrams we obtain a presentation of the singular pure virtual braid monoid.