Reorienting quandle orbits

TL;DR

The paper analyzes the orientation-reversal of a quandle orbit, a concept motivated by knot theory, and formalizes it as $Q^{\mathrm{rev}}(x)$ obtained by inverting $\beta_y$ for $y$ in the orbit $Q_x$. It introduces the notion $Q^{\mathrm{rev}}(x)_{\mathcal{V}}$ to test whether a quandle variety $\mathcal{V}$ preserves enough information to recover the original quandle, defining a variety as suited if this holds for all $Q$. The main result demonstrates that the medial-quandle variety $\mathcal{M}$ is not suited by constructing an infinite medial quandle $Q$ with two orbits for which $Q^{\mathrm{rev}}(0_2)_{\mathcal{M}}$ collapses to a two-element trivial quandle, showing that reversal plus a quotient can lose essential structure. This indicates limitations of naive orientation-reversals in medial settings and motivates developing orientation notions compatible with quandle-variety structure.

Abstract

Motivated by knot theory, it is natural to define the orientation-reversal of a quandle orbit by inverting all the translations given by elements of that orbit. In this short note we observe that this natural notion is unsuited to medial quandles.

Figures (1)

• Figure 1: The underpassing arc $b$ is on the right of $a$, and $b \triangleright a$ is on the left.

Theorems & Definitions (7)

• Definition 1
• Definition 2
• Lemma 3
• Lemma 4
• proof
• Theorem 5
• proof