On the ribbon graphs of links in real projective space

TL;DR

This paper explores the relationship between link diagrams in real projective space and their associated ribbon graphs, identifying moves that relate diagrams with identical ribbon graph representations.

Contribution

It characterizes how checkerboard colorable link diagrams in real projective space relate through ribbon graphs and introduces moves connecting diagrams with single-vertex ribbon graphs.

Findings

Identifies relations between link diagrams with the same ribbon graphs.

Defines moves that connect diagrams with one-vertex ribbon graphs.

Provides a framework for understanding link diagram equivalences in real projective space.

Abstract

Every link diagram can be represented as a signed ribbon graph. However, different link diagrams can be represented by the same ribbon graphs. We determine how checkerboard colourable diagrams of links in real projective space, and virtual link diagrams, that are represented by the same ribbon graphs are related to each other. We also find moves that relate the diagrams of links in real projective space that give rise to (all-A) ribbon graphs with exactly one vertex.