Virtually Symmetric representations and marked Gauss diagrams

Abstract

In this paper, we define the notion of a virtually symmetric representation of representations of virtual braid groups and prove that many known representations are equivalent to virtually symmetric. Using one such representation, we define the notion of virtual link groups which is an extension of virtual link groups defined by Kauffman. Moreover, we introduce the concept of marked Gauss diagrams as a generalisation of Gauss diagrams and their interpretation in terms of knot-like diagrams. We extend the definition of virtual link groups to marked Gauss diagrams and define their peripheral structure. We define $C_m$-groups and prove that every group presented by a $1$-irreducible $C_1$-presentation of deficiency $1$ or $2$ can be realized as the group of a marked Gauss diagram.

27 pages, 13 figures. Minor change in the structure of the paper, symbols, figures, accepted in Topology and its Applications

References in corpus (4)

• Alexander invariants for virtual knots
• On the welded Tube map
• Update of "Biquandles of Small Size and some Invariants of Virtual and Welded Knots"
• Spatial graph as connected sum of a planar graph and a braid