A relationship between multiple conjugation quandle/biquandle colorings

TL;DR

This paper establishes a functorial relationship between multiple conjugation biquandles and quandles, showing a correspondence between their colorings for handlebody-links and graphs, and proves an isomorphism for Alexander biquandle and quandle colorings.

Contribution

It introduces a functor from biquandles to quandles and demonstrates a diagrammatic correspondence between their colorings, including an isomorphism for Alexander structures.

Findings

One-to-one correspondence between biquandle and quandle colorings.

Diagrammatic equivalence for handlebody-link and graph colorings.

Isomorphism of Alexander biquandle and quandle coloring modules.

Abstract

We define a functor $\mathcal{Q}$ from the category of multiple conjugation biquandles to that of multiple conjugation quandles. We show that for any multiple conjugation biquandle $X$, there is a one-to-one correspondence between the set of $X$-colorings and that of $\mathcal{Q}(X)$-colorings diagrammatically for any handlebody-link and spatial trivalent graph. In particular, we prove that the set of $G$-family of Alexander biquandles colorings is isomorphic to that of $G$-family of Alexander quandles colorings as modules.