Abstract
This paper is an exploration of simple four-regular graphs in the plane (i.e. loop-free and with no more than one edge between any two nodes). Such graphs are fundamental to the theory of knots and links in three dimensional space, and their planar diagrams. We dedicate this paper to Frank Harary (1921–2005), whose fascination with graphs of knots inspired this work, and with whom we had the pleasure of developing this paper.
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