Polynomial Generalizations of Knot Colorings

  • Rachael He University of Rochester
  • Austin Ho University of Rochester
  • Dorian Kalir University of Rochester
  • Jacob Miller University of Rochester
  • Matthew Zevenbergen University of Rochester
Keywords: knot theory; knot invariants; Reidemeister moves


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.

How to Cite
He, R., Ho, A., Kalir, D., Miller, J., & Zevenbergen, M. (2022). Polynomial Generalizations of Knot Colorings. The PUMP Journal of Undergraduate Research, 5, 1-23. Retrieved from https://journals.calstate.edu/pump/article/view/2616