Virtual braids and virtual curve diagrams

There is a well known injective homomorphism $\phi:{\mathcal {B}}_n \rightarrow {\rm Aut}(F_n)$fromtheclassicalbraidgroup${\mathcal {B}}_n$intotheautomorphismgroupofthefreegroup$F_n$, 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 ${\mathcal{V}\mathcal{B}}_n$ of virtual braids. In this paper, we will generalize the above action to ${\mathcal{V}\mathcal{B}}_n$. We will define a set, ${\mathcal {VCD}}_n$,of\mathrm{"}virtualcurvediagrams\mathrm{"}anddefineanactionof${\mathcal {VB}}_n$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 ${\mathcal{V}\mathcal{B}}_n$. 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 $n=4$.