Asymmetric Hyperbolic Knots Detect Mapping Classes of 3-Manifolds: A Written Proof for a Problem of the K3 List

Problem 1.19 of the list K3: A New Problem List in Low-Dimensional Topology (Baykur, Kirby and Ruberman, eds., 2026) asks: if Y = Y₁ # Y₂ is a connected sum of 3-manifolds, neither of them S³, and the twist Φ about the connected-sum sphere is not isotopic to the identity, is there a knot K ⊂ Y such that K and Φ(K) are not isotopic? The question goes back to Aceto, Bregman, Davis, Park and Ray. They proved the analogous statement for prime 3-manifolds, conjectured it in general, and recorded, in all versions of their paper since 2020, that Etnyre and Margalit have a proof of the general statement. That proof has not appeared yet; their paper is forthcoming, and to our knowledge no written proof exists. This note supplies one, for closed orientable 3-manifolds. According to D. Margalit (personal communication) the argument given here has no relation to theirs. We claim no priority for the result. Let K be a knot in a closed, connected, orientable 3-manifold Y whose complement is hyperbolic with trivial isometry group; such knots exist by a theorem of Kawauchi. We show that every diffeomorphism f of Y for which f(K) is isotopic to K is isotopic to the identity. Hence every diffeomorphism that is not isotopic to the identity changes the isotopy class of a knot, and the answer to Problem 1.19 is yes. The proof is the classical description of the symmetries of a hyperbolic knot (Mostow–Prasad, Waldhausen, Hatcher). Using a theorem of Chen and Tshishiku on finite group actions, we also show that a twist about a connected-sum sphere which is not isotopic to the identity changes the isotopy class of every knot with hyperbolic complement. This is an unrefereed note. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Version 1.1 (5 October 2026): the statements about the relation of this note to the proof of Etnyre and Margalit were updated after a personal communication of D. Margalit; the mathematical content is unchanged. Corpus identifier: KP-1.19 (R. İ. Baykur, R. C. Kirby, D. Ruberman, eds., K3: A New Problem List in Low-Dimensional Topology, AMS Mathematical Surveys and Monographs 295, 2026, Problem 1.19).