Knots and Graphs I—Arc Graphs and Colorings

Abstract

It is well-known that combinatorial knot theory is a graph theoretic way to approach the theory of knots, links, and their topology in three-dimensional space. The usual formulation of this theory via knot diagrams works with a formal system that is a mixture of diagrams and graphs. Our Ž purpose is to introduce a specific, clear and simple graph theoretic and . hence set-theoretic foundation for the theory of knot and link diagrams. In the course of our work, we introduce a very natural arc-graph associated with any knot or link diagram, and we show how these arc-graphs can be put to use in studying the algebraic invariants of knots such as the fundamental group, the quandle or the involutory quandle. By specializing Ž the quandle to the case of colorings with values in ZrnZ the integers . Ž . modulo n , we define a new invariant, the coloring number C K , for knots and links. There are two main variations of this coloring number, Ž . Ž . Ž . that we denote by C K and CM K . CM K is the least number of colors needed on a minimal diagram for K. We close with a discussion about this new invariant and conjectures related to it. Section 2 sets the background for knot and link diagrams, and defines Ž . the arc-graph A K of a diagram K. We then define admissible labellings Ž . G, L for plane 4-regular graphs G and show that such labelled graphs