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There is a well-known way to describe a link diagram as a (signed) planegraph, called its Tait graph. This concept was recently extended, providing away 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, notevery embedded graph represents a link diagram. Furthermore, although a Taitgraph describes a unique link diagram, the same embedded graph can representmany different link diagrams. One is then led to ask which embedded graphsrepresent link diagrams, and how link diagrams presented by the same embeddedgraphs are related to one another. Here we answer these questions bycharacterizing the class of embedded graphs that represent link diagrams, andthen using this characterization to find a move that relates all of the linkdiagrams that are presented by the same set of embedded graphs.

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