Affine G-multibiquandle modules for colored links I: representability of coloring functors and color-profile polynomials
Abstract
Let R be a commutative ring with identity, let G be an abelian group, and fix u∈ R[G]×. For each component-labeled oriented link diagram (D,σ), we construct a finitely presented R[G]-module Mu(D,σ;R[G]) with a universal affine coloring UD,σ. Every affine coloring with values in an R[G]-module factors uniquely through UD,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.
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BibTeX
@misc{lee2026affine,
title = {{Affine $G$-multibiquandle modules for colored links I: representability of coloring functors and color-profile polynomials}},
author = {Sang Youl Lee},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29011},
eprint = {2610.29011},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29011},
}
RIS
TY - UNPB
TI - Affine G-multibiquandle modules for colored links I: representability of coloring functors and color-profile polynomials
AU - Sang Youl Lee
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29011
UR - https://arxiv.org/abs/2610.29011
AB - Let R be a commutative ring with identity, let G be an abelian group, and fix u∈ R[G]^×. 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.
ER -