Knots and Eulerian Cycles

This chapter provides an introduction to some of the graph theory associated with knots and links. The connection arises from the description of the shadow of a link diagram as a 4-valent plane graph. The link diagram is determined by a particular eulerian tour in this graph, and consequently many operations on link diagrams translate to operations on eulerian tours in plane graphs. The study of eulerian tours in 4-valent plane graphs leads naturally to the study of a number of interesting combinatorial objects, such as double occurrence words, chord diagrams, circle graphs, and maps. Questions that are motivated by the theory of knots and links can often be clarified or solved by being reformulated as a question in one of these different contexts.