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

Iain Moffatt

Algebraic & Geometric Topology 12 (2012) 1099–1136

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.

$1$–sum, checkerboard graph, dual, embedded graph, knots and links, Partial duality, plane graph, ribbon graph, separability, Tait graph, Turaev surface
Mathematical Subject Classification 2010
Primary: 05C10, 57M15
Secondary: 57M25, 05C75
Received: 10 January 2012
Revised: 23 February 2012
Accepted: 25 February 2012
Published: 19 May 2012
Iain Moffatt
Department of Mathematics and Statistics
University of South Alabama
411 University Blvd N
Mobile AL 36688