  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.
