Two-Variable Parity Polynomial for Virtual Knotoids

In this paper, we introduce a two-variable parity polynomial invariant for virtual knotoids, defined on oriented virtual knotoid diagrams. The construction is based on the parity of classical crossings, where each crossing is classified as even or odd and treated accordingly in the definition of the invariant. We study several fundamental properties of this invariant. We demonstrate that the parity polynomial can distinguish pairs of virtual knotoids that are not distinguished by the odd writhe, prove that it is equivalent to the affine index polynomial, and establish that it is a Vassiliev invariant of order one. Finally, we give its relationship with the Petit gluing invariant.