The virtual and universal braids

We study the structure of the virtual braid group. It is shown that the virtual braid group is a semi--direct product of the virtual pure braid group and the symmetric group. Also, it is shown that the virtual pure braid group is a semi--direct product of free groups. From these results we obtain a normal form of words in the virtual braid group. We introduce the concept of a universal braid group. This group contains the classical braid group and has as its quotient groups the singular braid group, virtual braid group, welded braid group, and classical braid group.Comment: 20 pages, 5 figure

On certain questions of the free group automorphisms theory

Certain subgroups of the groups $Aut(F_n)$ of automorphisms of a free group $F_n$ are considered. Comparing Alexander polynomials of two poly-free groups $Cb_4^+$ and $P_4$ we prove that these groups are not isomorphic, despite the fact that they have a lot of common properties. This answers the question of Cohen-Pakianathan-Vershinin-Wu from \cite{CVW}. The questions of linearity of subgroups of $Aut(F_n)$ are considered. As an application of the properties of poison groups in the sense of Formanek and Procesi, we show that the groups of the type $Aut(G*\mathbb Z)$ for certain groups $G$ and the subgroup of $IA$-automorphisms $IA(F_n)\subset Aut(F_n)$ are not linear for $n\geq 3$. This generalizes the recent result of Pettet that $IA(F_n)$ are not linear for $n\geq 5$.Comment: 11 page

Groups of virtual and welded links

We define new notions of groups of virtual and welded knots (or links) and we study their relations with other invariants, in particular the Kauffman group of a virtual knot.Comment: 22 pages, 7 figure