Affine $G$-multibiquandle modules for colored links I: representability of coloring functors and color-profile polynomials

Source abstract

Let $R$ be a commutative ring with identity, let $G$ be an abelian group, and fix $u\in R[G]^\times$. For each component-labeled oriented link diagram $(D,σ)$, we construct a finitely presented $R[G]$-module $M_u(D,σ;R[G])$ with a universal affine coloring $U_{D,σ}$. Every affine coloring with values in an $R[G]$-module factors uniquely through $U_{D,σ}$ by a module homomorphism. Thus the module represents the affine coloring functor. Each specified colored Reidemeister move induces a unique module isomorphism compatible with coloring transport, and the module isomorphism class is a colored-link invariant. The representing modules and universal colorings commute with arbitrary scalar extension, without a flatness assumption. For classical links, simultaneous rescaling of the component labels removes the common deformation unit. For any multibiquandle, associated-biquandle colorings decompose into fibers indexed by component labelings. For finite targets and palettes, we define a color-profile polynomial recording the joint distribution of fiber sizes and numbers of palette labels used, and prove its invariance under oriented link isotopy. In the affine case, the representing modules determine these fibers, and their specialized dimensions determine the coloring counts over finite fields. The color-profile polynomial is not determined, in general, by the full unweighted distribution of coloring-fiber sizes. It therefore retains information beyond the associated-biquandle coloring count.