Partial duals of plane graphs, separability and the graphs of knots

TL;DR

This paper characterizes which embedded graphs can represent link diagrams and introduces a move that relates all link diagrams sharing the same embedded graph set, advancing understanding of graph-link diagram relationships.

Contribution

It provides a characterization of embedded graphs representing link diagrams and identifies a move connecting all diagrams derived from the same embedded graphs.

Findings

Characterization of embedded graphs representing link diagrams

Identification of a move relating all diagrams from the same embedded graphs

Insight into the relationship between link diagrams and their associated graphs

Abstract

There is a well-known way to describe a link diagram as a (signed) plane graph, called its Tait graph. This concept was recently extended, providing a way to associate a set of embedded graphs (or ribbon graphs) to a link diagram. While every plane graph arises as a Tait graph of a unique link diagram, not every embedded graph represents a link diagram. Furthermore, although a Tait graph describes a unique link diagram, the same embedded graph can represent many different link diagrams. One is then led to ask which embedded graphs represent link diagrams, and how link diagrams presented by the same embedded graphs are related to one another. Here we answer these questions by characterizing the class of embedded graphs that represent link diagrams, and then using this characterization to find a move that relates all of the link diagrams that are presented by the same set of embedded graphs.