Virtually Symmetric representations and marked Gauss diagrams

Abstract

In this paper we define virtually symmetric representations of virtual braid group V Bn and prove that many previously known representations are equivalent to virtually symmetric representations. Using a virtually symmetric representation we define virtual link group. We define and study group system of virtual knots by defining marked Gauss diagrams as an extension of Gauss diagrams. A marked Gauss diagram is a Gauss diagram with marked nodes (signed vertices) on circles, and are not attached to arrows. In particular, we extend the definition of virtual link group to marked Gauss diagrams and define peripheral structure for 1-circle marked Gauss diagrams. We define Cm-groups and prove that every irreducible C1-group can be realized as the group of a marked Gauss diagram. We give an interpretation of marked Gauss diagrams in terms of virtual spatial graph diagrams with marked nodes. We also study peripherally specified homomorphic image of groups associated to marked Gauss diagrams.