{
  "schema_version": 1,
  "problem_number": "KP-1.19",
  "title": "Asymmetric Hyperbolic Knots Detect Mapping Classes of 3-Manifolds: A Written Proof for a Problem of the K3 List",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "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.",
  "result_type": "COMPLETE_SCOPED_PROOF",
  "categories": [
    "math.GT"
  ],
  "keywords": [
    "knots in 3-manifolds",
    "isotopy of knots",
    "equivalent knots",
    "mapping class group of a 3-manifold",
    "sphere twist",
    "Gluck twist",
    "connected sum",
    "hyperbolic knot",
    "asymmetric knot",
    "Mostow rigidity",
    "K3 problem list",
    "Problem 1.19",
    "UnsolvedMath",
    "KP-1.19",
    "math.GT",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-03",
  "publication_date": "2026-10-03",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-03",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/kp-1-19/",
  "pdf_url": "https://eulersolve.org/papers/kp-1-19/paper.pdf?v=eb4a16c16f46",
  "doi": "10.5281/zenodo.23127199",
  "zenodo_record_url": "https://zenodo.org/records/23127199",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Answers Problem 1.19 of the K3 problem list (2026) affirmatively for closed, connected, orientable 3-manifolds in the smooth category, and proves the conjecture of Aceto, Bregman, Davis, Park and Ray in that setting. All versions of their paper since 2020 record that Etnyre and Margalit have a proof of the general statement, which has not appeared. This note supplies a written proof; it may coincide with their argument, claims no priority, and was written without contact with them. The ingredients are known theorems (Kawauchi, Mostow–Prasad, Waldhausen, Cerf, Hatcher, Chen–Tshishiku). Unrefereed.",
  "files": {
    "paper.pdf": {
      "sha256": "eb4a16c16f46ab870e82e582793703216c13def802c5f3870a02b4ea4ec83bf7"
    },
    "source.zip": {
      "sha256": "64715665496aef87b5ea0c1f50698c4d68884edc3bde88a6930361b250a1452c"
    },
    "verification_report.md": {
      "sha256": "7c2a2d103b8b311c518f3de098a163a5360f8f7babfd2b507ee2b19232ba7e11"
    }
  },
  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}