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

Abstract

In the paper, we construct a representation [Formula: see text] of the flat virtual braid group [Formula: see text] on [Formula: see text] strands by automorphisms of the free group [Formula: see text] with [Formula: see text] 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 [Formula: see text] in [Formula: see text], [Formula: see text] in [Formula: see text], [Formula: see text] in [Formula: see text], which play an important role in the study of the kernel of the representation [Formula: see text].