From Fox 3-Coloring to the Yang-Baxter Operator and Its Homology

This lecture is divided into two parts. The first part of the lecture is more elementary and explores the Fox 3-coloring invariant and its generalization into quandle invariants. Here we also describe the Wirtinger’s and Dehn’s presentations of the fundamental group of the link complement and their relation to Fox colorings. The second part of the lecture describes how the idea of Fox colorings can develop into the sophisticated notion of Yang-Baxter operator and its homology. The Yang-Baxter equation was extensively studied by Chen N. Yang in 1968 and by Rodney J. Baxter in 1971. The relation between the equation and knot theory was first noted by Vaughan F. R. Jones, when he constructed the Yang-Baxter operator yielding his polynomial and the HOMFLYPT polynomial. Yang-Baxter homology in full generality was introduced by Victoria Lebed and the first author in 2012. Since then, it has become of substantial interest to knot theorists.