Projection Image Polynomial Invariants for Dichromatic Links with Dibiquandle Colorings

Abstract

We introduce a new polynomial invariant for oriented dichromatic links, called the projection image polynomial, derived from the dibiquandle colorings of link diagrams. This invariant simultaneously extends the dibiquandle counting invariant and the image enhancement polynomial by incorporating the two independent biquandle projections of a dibiquandle. We establish that the projection image polynomial is preserved under oriented dichromatic Reidemeister moves and provide an explicit correspondence between colorings and homomorphisms of the fundamental dibiquandle. Through several examples, we show that this polynomial distinguishes dichromatic links which share identical counting invariants and image enhancement polynomial. Furthermore, we prove its algebraic properties, including multiplicativity under disjoint union, a normalization law for connected sum, and duality relations under mirror image and orientation reversal. Finally, we outline a natural generalization of this framework to oriented n–colored links via the notion of an n–multibiquandle, which leads to the associated n–projection image polynomial. This multicolored extension unifies the biquandle (n = 1) and dibiquandle (n = 2) cases within a common algebraic framework and provides a foundation for subsequent module–theoretic and categorified refinements.