Representations of Flat Virtual Braids by Automorphisms of Free Group

Representations of braid group Bn on n >= 2 strands by automorphisms of a free group of rank n go back to Artin. In 1991, Kauffman introduced a theory of virtual braids, virtual knots, and links. The virtual braid group VBn on n >= 2 strands is an extension of the classical braid group Bn by the symmetric group Sn. In this paper, we consider flat virtual braid groups FVBn on n >= 2 strands and construct a family of representations of FVBn by automorphisms of free groups of rank 2n. It has been established that these representations do not preserve the forbidden relations between classical and virtual generators. We investigated some algebraic properties of the constructed representations. In particular, we established conditions of faithfulness in case n=2 and proved that the kernel contains a free group of rank two for n >= 3.

Automated summary

This study examines the representations of the flat virtual braid group FVBn on n>=2 strands, and establishes that these representations do not preserve the forbidden relations between classical and virtual generators. It also proves the conditions of faithfulness for these representations and discovers that the kernel contains a free group of rank two for n >= 3.