Abstract:

 In the present paper, we study various representations of the virtual braid group $VB_n$, the singular braid monoid $SM_n$, and the singular braid group $SB_n$. We construct  a representation of the virtual braid group into the group of automorphisms of the quandle. We prove that this representation generalizes some well-known representations.  Using this representation, we construct an invariant of virtual links.
Also we investigate  representations of the singular braid monoid $SM_n$ and the singular braid group $SB_n$ by endomorphisms and automorphisms of a free module and a free group. In particular, we find all linear local homogeneous representations of the singular braid monoid that extend the Burau representation of the braid group and find conditions under which these representations can be extended to the group $SB_n$. We prove that the constructed linear representation is reducible and an irreducible representation is obtained that extends the reduced Burau representation.