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, and to our knowledge no written proof exists. This note supplies one, for closed orientable 3-manifolds. It may well coincide with the argument of Etnyre and Margalit, with whom we have not been in contact, and we claim no priority. 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.