Polynomial Generalizations of Knot Colorings

Authors

  • 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

DOI:

https://doi.org/10.46787/pump.v5i0.2616

Keywords:

knot theory; knot invariants; Reidemeister moves

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.

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Published

2022-01-01

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. https://doi.org/10.46787/pump.v5i0.2616