  In this paper, we define some polynomial invariants for virtual knots and
links. In the first part we use Manturov's parity axioms to obtain a new
polynomial invariant of virtual knots. This invariant can be regarded as a
generalization of the odd writhe polynomial defined by the first author. The
relation between this new polynomial invariant and the affine index polynomial
is discussed. In the second part we introduce a polynomial invariant for long
flat virtual knots. In the third part we define a polynomial invariant for
2-component virtual links. This polynomial invariant can be regarded as a
generalization of the linking number.
