Analysis reveals that a continuous map connects quandle orders with actions on one-manifolds, suggesting new pathways for studying rigidity.
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.
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Jibiki et al. (2025) studied this question.
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