Quandles, n-Colorability, and Cocycles

Abstract

In the 1960s Ralph Fox introduced a tool of knot theory called colorability. Briefly, a knot is n-colorable if the diagram for that knot can be decorated with numbers 1, 2, . . . , n subject to a simple constraint at each crossing. Whether or not a diagram of a knot admits an n-coloring turns out to be independent of the diagram. Thus, one can attempt to distinguish knots by asking if they admit n-colorable diagrams. In this project, we study n-colorings of n-colorable knots. In particular if one colors a single knot diagram in two different ways then when can these seemingly distinct colorings be said to be deformed into each other? On the way, we use tools from the theory of quandles including Alexander quandles and cocycle enhancements.