Virtually Symmetric representations and marked Gauss diagrams

TL;DR

This paper introduces virtually symmetric representations for virtual braid groups, extends virtual link groups via marked Gauss diagrams, and demonstrates their realizability for certain group presentations, enriching the algebraic and diagrammatic tools in virtual knot theory.

Contribution

It defines virtually symmetric representations, introduces marked Gauss diagrams, and extends virtual link groups, providing new algebraic and diagrammatic frameworks in virtual knot theory.

Findings

Many known representations are equivalent to virtually symmetric.

Virtual link groups are extended using marked Gauss diagrams.

Groups with specific presentations can be realized as marked Gauss diagram groups.

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.