  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.
