Virtual braids and virtual curve diagrams

TL;DR

This paper extends the classical braid group action to virtual braids using virtual curve diagrams, proving faithfulness and providing a combinatorial solution to the word problem, while also showing the non-injectivity of a related homomorphism.

Contribution

It introduces virtual curve diagrams and an action of virtual braid groups, proving faithfulness and addressing the injectivity of a known homomorphism.

Findings

The action of virtual braid groups on virtual curve diagrams is faithful.

A combinatorial solution to the word problem in virtual braid groups is provided.

The homomorphism from virtual braid groups to automorphisms of free groups is not injective for n=4.

Abstract

There is a well known injective homomorphism $\phi:{\mathcal {B}}_n \rightarrow {\rm Aut}(F_n)$ from the classical braid group ${\mathcal {B}}_n$ into the automorphism group of the free group $F_n$, first described by Artin. This homomorphism induces an action of ${\mathcal {B}}_n$ on $F_n$ that can be recovered by considering the braid group as the mapping class group of $H_n$ (an upper half plane with $n$ punctures) acting naturally on the fundamental group of $H_n$. Kauffman introduced virtual links as an extension of the classical notion of a link in ${\mathbb {R}}^3$. As in the classical case, there is a corresponding group ${\mathcal {VB}}_n$ of virtual braids. In this paper, we will generalize the above action to ${\mathcal {VB}}_n$. We will define a set, ${\mathcal {VCD}}_n$, of "virtual curve diagrams" and define an action of ${\mathcal {VB}}_n$ on ${\mathcal {VCD}}_n$. Then, we will show that, as in Artin's case, the action is faithful. This provides a combinatorial solution to the word problem in ${\mathcal {VB}}_n$. Bardakov and Manturov described an extension $\psi:{\mathcal {VB}}_n\rightarrow {\rm Aut}(F_{n+1})$ of the Artin homomorphism, and raised the question of its injectivity. We find that $\psi$ is not injective by exhibiting a non-trivial virtual braid in the kernel when $n=4$.