Virtually symmetric representations and marked Gauss diagrams

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

“…Therefore θ preserves all defining relations (1)-( 7), (10) of F V B n , and hence θ induces the representation θ :…”

“…Proof. In order to show that the map θ given by ( 14) induces the representation of the flat virtual braid group F V B n , it is sufficient to show that θ preserves all defining relations (1)-( 7), (10) of the flat virtual braid group F V B n . For i = 1, 2, .…”

“…, n − 1 denote by s i , r i ∈ Aut(F 2n ) respectively the images of the elements σ i , ρ i ∈ F V B n under the map θ. The fact that θ preserves relations (3), (10) means that the equalities r 2 i = id, s 2 i = id hold for all i = 1, 2, . .…”

“…In order to find conditions under which the mapping Θ gives a representations, we have to check when the mapping Θ preserves the relations ( 1)-( 7), (10) of the flat virtual braid group F V B n . Using direct calculations it is easy to see that the mapping Θ preserves relations ( 1)-( 7), (10) if and only if p 1 = p 2 = • • • = p n (we denote all of these elements by p), q 1 = q 2 = • • • = q n = 1. This fact implies the following theorem.…”

“…Recently in the paper [10] the notion of the virtually symmetric representation of braid-like groups were defined. Theorem 6 and the definition of the representation θ given in Section 3 say that all representations given by formulas ( 21)-( 24) are virtually symmetric.…”