Dynamics of Quandle Orders

The paper proves the equivalence between isolated circular orders on the Dehn quandle D(G,A) and strong rigidity of the associated circle action, under the semi-latin hypothesis and an extension of the dynamical realization to a G-action (Theorem 1.2). The proof hinges on (i) reconstructing the order from the action via the index λ at the chosen basepoint (Definition 4.1 and Remark 4.2), (ii) a semiconjugacy criterion that requires the basepoint to have trivial stabilizer (Lemma 4.4), and (iii) a one-sided continuity statement of the realization map (Lemma 4.6), with a case split using the exceptional minimal set alternative (Lemma 4.5). These ingredients are all present and used coherently in the paper . By contrast, the candidate solution omits the index λ entirely, asserts that any quandle action gives a circular order by reading off the orbit without handling stabilizers, and misuses continuity of the realization map as if it were locally surjective onto neighborhoods of actions. It also fails to treat the crucial ‘nontrivial stabilizer’ case that the paper handles to complete (1 ⇒ 2) (see the proof of Theorem 1.2) .