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

Moffatt, Iain

doi:10.2140/agt.2012.12.1099

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

Iain Moffatt

Algebraic & Geometric Topology 12 (2012) 1099–1136

DOI: 10.2140/agt.2012.12.1099

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.

Keywords

$1$–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

References

Publication

Received: 10 January 2012

Revised: 23 February 2012

Accepted: 25 February 2012

Published: 19 May 2012

Authors

Iain Moffatt
Department of Mathematics and Statistics University of South Alabama 411 University Blvd N Mobile AL 36688 USA
http://www.southalabama.edu/mathstat/personal\_pages/imoffatt/

Volume 12, issue 2 (2012)