The fundamental multiple conjugation quandle of a handlebody-link

Abstract

A handlebody-link is a disjoint union of handlebodies embedded in the 3-sphere $S^3$. A multiple conjugation quandle is an algebraic system whose axioms are derived from the Reidemeister moves for handlebody-link diagrams. In this paper, we introduce the notion of a presentation of a multiple conjugation quandle and define the fundamental multiple conjugation quandle of a handlebody-link. We also see that the fundamental multiple conjugation quandle is an invariant of handlebody-links.