Three Dimensions of Knot Coloring

A knot is a circle smoothly embedded in 3-dimensional Euclidean space or its compactification, the 3-sphere. Two knots are regarded as the same if one can be smoothly deformed into the other.1 The mathematical theory of knots emerged from the smoky ruins of Lord Kelvin’s “vortex atom theory,” a hopelessly optimistic theory of matter of the nineteenth century in which atoms appeared as microscopic vortices of æther. Kelvin was inspired by theorems of Hermann von Helmholtz on vortex motion as well as poisonous smoke-ring laboratory demonstrations of a fellow Scot, Peter Guthrie Tait. (See [13] for a historical account.) More than anyone else, Tait recognized the mathematical profundity of the nascent subject. He was the author of the first publication with the word “knot” in its title. As Tait knew, a knot can be represented by a diagram, a regular 4valent graph in the plane with a tromp l’oeil device at each vertex indicating how one arc passes over another, the “hidden line” device that has been universally adopted today. Homeomorphisms of the plane might distort the graph, but they do not change the knot. Going deeper, a theorem of Kurt Reidemeister from 1926 (also proved independently by J.W. Alexander and his student G.B. Briggs one year later) informs us that two diagrams represent the same knot if and only if one can be converted into the other by a finite sequence of local changes, today called Reidemeister moves (see, for example, [4] [10]).