Representations of flat virtual braids which do not preserve the forbidden relations

Abstract

In the paper, we construct a representation $𝜃:{\mathrm{\text{FVB}}}_n\to \mathrm{\text{Aut}}(F_{2n})$ of the flat virtual braid group ${\mathrm{\text{FVB}}}_n$ on $n$ strands by automorphisms of the free group $F_{2n}$ with $2n$ generators which does not preserve the forbidden relations in the flat virtual braid group. This representation gives a positive answer to the problem formulated by Bardakov in the list of unsolved problems in virtual knot theory and combinatorial knot theory by Fenn et al.

Also we find the set of normal generators of the groups ${\mathrm{\text{VP}}}_n\cap H_n$ in ${\mathrm{\text{VB}}}_n$, ${\mathrm{\text{FVP}}}_n\cap {\mathrm{\text{FH}}}_n$ in ${\mathrm{\text{FVB}}}_n$, ${\mathrm{\text{GVP}}}_n\cap {\mathrm{\text{GH}}}_n$ in ${\mathrm{\text{GVB}}}_n$, which play an important role in the study of the kernel of the representation $𝜃$.