Algebraic, combinatorial and topological properties of singular virtual braid monoids

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 singular virtual 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, singular horizontal Gauss diagrams up to a certain equivalence relation, and singular virtual braids, in particular using singular horizontal Gauss diagrams we obtain a presentation of the singular pure virtual braid monoid.