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

TL;DR

This paper constructs a novel representation of the flat virtual braid group that does not preserve forbidden relations, providing new insights and invariants in virtual knot theory.

Contribution

It introduces a new representation of the flat virtual braid group that answers an open problem and leads to a new invariant for flat welded links.

Findings

Constructed a representation $ heta$ of $FVB_n$ on automorphisms of $F_{2n}$

Provided a positive solution to an open problem in virtual knot theory

Identified generators of specific subgroups related to the kernel of the representation

Abstract

In the paper, we construct a representation $\theta:FVB_n\to{\rm Aut}(F_{2n})$ of the flat virtual braid group $FVB_n$ 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 V. Bardakov in the list of unsolved problems in virtual knot theory and combinatorial knot theory by R. Fenn, D. Ilyutko, L. Kauffman and V. Manturov. Using this representation we construct a new group invariant for flat welded links. Also we find the set of normal generators of the groups $VP_n\cap H_n$ in $VB_n$, $FVP_n\cap FH_n$ in $FVB_n$, $GVP_n\cap GH_n$ in $GVB_n$, which play an important role in the study of the kernel of the representation $\theta$.