An Index-Graded Virtual Intersection Polynomial for Oriented Virtual Knots

Polynomial invariants are important tools for studying virtual knots, and different constructions encode different aspects of the combinatorial and geometric information of a virtual knot diagram. Motivated by the interaction between crossing indices and smoothing operations, we construct a bivariate polynomial invariant PD(x,t) for oriented virtual knots. The construction combines the affine index, the absolute virtual intersection index, and the change of an auxiliary polynomial under a prescribed smoothing that produces one component. We prove that PD(x,t) is invariant under all oriented generalized Reidemeister moves and determine its behavior under orientation reversal and mirror image. We construct an infinite family for which the affine index polynomial, the virtual intersection polynomial PD(t), and the auxiliary polynomial QD(t) all vanish, whereas PD(x,t) distinguishes all members of the family. A second example shows that the variable x retains information that is lost after the specialization x=1. Moreover, using positive and negative resolutions of singular crossings, we prove that PD(x,t) has Vassiliev degree exactly one. These results provide a way to organize smoothing information according to the virtual intersection indices of the original crossings.