Dynamics of quandle orders

TL;DR

This paper constructs a continuous map linking quandle orders to actions on one-manifolds, characterizes isolated orders via rigidity, and shows the rarity of isolated right orders on free quandles.

Contribution

It introduces a novel continuous map from quandle orders to actions on manifolds and characterizes isolated orders in this context.

Findings

Characterized isolated orders on quandles using strong rigidity.

Proved non-existence of isolated right orders on free quandles in most cases.

Provided a new answer to a question by Ba and Elhamdadi.

Abstract

We construct a continuous map from the space of orders on quandles to the space of quandle actions on one-manifolds, providing an answer to a question posed by Idrissa Ba and Mohamed Elhamdadi. As an application of this map, we characterize isolated orders on quandles in terms of strong rigidity. As another application, we prove that there is no isolated right order on the free quandles, except for specific cases.