The intersection polynomials of a virtual knot I: Definitions and calculations

Abstract

We introduce three kinds of invariants of a virtual knot called the first, second, and third intersection polynomials. The definition is based on the intersection number of a pair of curves on a closed surface. The calculations of intersection polynomials are given up to crossing number four. We also study several properties of intersection polynomials.

27 pages, 22 figures. We divide the original version into four papers