Polynomial Generalizations of Knot Colorings

Abstract

In the field of knot theory, knot invariants are properties preserved across all embeddings and projections of the same knot. Fox n-coloring is a classical knot invariant which associates to each knot projection a system of linear equations. We generalize Fox’s n-coloring by using two, not necessarily distinct, polynomials over a field F, which we say form a (g,f)_F coloring. We introduce a sufficient condition, called strong, for a pair of polynomials to form a  (g,f)_F coloring. We confirm a family of pairs of linear polynomials each of which form a (g,f)_F coloring. We prove that there are no strong pairs containing an irreducible quadratic polynomial over a field F not of characteristic two. Furthermore, we find a method to produce polynomials with unbounded degree that form colorings over suitable fields.