The intersection polynomials of a virtual knot

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. We study properties of the intersection polynomials and their applications concerning the behavior on symmetry, the crossing number and the virtual crossing number, a connected sum of virtual knots, characterizations of intersection polynomials, finite type invariants of order two, and a flat virtual knot.