On Core Quandles

TL;DR

This work develops a group-theoretic framework for core quandles, revealing how core structures $\mathsf{Core}(G)$ reflect underlying group properties. Through two canonical homogeneous representations and a detailed analysis of the displacement group, it characterizes connectedness, simplicity, and primitivity of cores, and links many quandle-theoretic notions (faithfulness, latinness, 2-Engel, medial) to concrete group properties (centers, involutions, nilpotency, and Engel conditions). A comprehensive commutator-theory for cores is established, together with adjunctions between the Core functor and a universal adjoint $\mathcal{H}$, yielding practical criteria for embedding and colorings. The results also connect knot colorings by core quandles to colorings by faithful and affine substructures, illustrating the utility of cores as a universal medium for finite quandle colorings and knot invariants. Overall, the paper provides a unified, repository-like linkage between group-theoretic properties and core-quandle structure, with concrete consequences for representations, embeddings, and knot theory.

Abstract

We characterize several properties of core quandles in terms of the properties of their underlying groups. Specifically, we characterize connected cores providing an answer to an open question in \cite{saito} and present a standard homogeneous representation for them, which allows us to prove that simple core quandles are primitive.

Figures (1)

• Figure 1: Crossing relation.

Theorems & Definitions (72)

• Example 1.1
• Proposition 1.2: hsv, Proposition 3.5
• Proposition 1.3: hsv, Theorem 4.1, and J
• Example 1.4
• Remark 1.5
• Proposition 1.6
• proof
• Example 1.7
• Lemma 2.1
• proof