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

Abstract

In the paper we construct a representation θ : F V Bn → Aut(F2n) of the flat virtual braid group F V Bn on n strands by automorphisms of the free group F2n 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 V Pn ∩ Hn in V Bn, F V Pn ∩ F Hn in F V Bn, GV Pn ∩ GHn in GV Bn, which play an important role in the study of the kernel of the representation θ.

Representations of Flat Virtual Braids by Automorphisms of Free Group

“…In Theorem 4, it will be established that for n ≥ 3 the kernel of the homomorphism Θ n contains a free group of rank 2. We note it was shown earlier in [46] that for n ≥ 3 the kernel of the homomorphism η n , which is a special case of Θ n , contains an infinite cyclic group. In Section 5, we present a family of local non-homogeneous representations, see Theorem 5.…”

“…The following property is a generalization of the property established in [46] (Prop. 9) for the word w(A, B) = B. Lemma 2.…”

“…In [45] the following problem was formulated: Does it exist a representation of the FVB n group by automorphisms of some group for which the forbidden relations would not hold? In [46], such representation η n : FVB n → Aut(F 2n ) was constructed, here F 2n = x 1 , . .…”

“…Here, we use the term flat virtual kure braid since the term kure virtual braid group was used in [47] for kernel of the map π K : VB n → S n which is defined analogously to ν n : FVB n → S . The group FVK n = Ker(ν n ) also was denoted by FH n in [46] since it is the flat analog of the Rabenda's group H n from [48] (Prop. 17).…”

Recurrent Generalization of F-Polynomials for Virtual Knots and Links

“…F-polynomials were calculated for tabulated virtual knots in [8] and [19], and successfully used to distinguish some oriented virtual knots in [6]. Another approach to construct invariants of flat virtual knots can be based on representation of flat virtual braids by automorphisms of free groups, see, for example, [1].…”