{
  "schema_version": 1,
  "problem_number": "OWR-14299580-006",
  "title": "Drinfeld Twists Between Non-Isomorphic Rack Solutions: A Negative Answer to a Question of Ferri",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "In an extended abstract in Oberwolfach Report 49/2025 (Oberwolfach Rep. 22 (2025), no. 4, 2633–2672; pp. 2645–2649), D. Ferri proposes to classify set-theoretic solutions of the Yang–Baxter equation up to Drinfeld twists, and asks whether two rack solutions on the same set that are related by a Drinfeld twist must coincide. Isomorphic solutions are always related by a twist (Ghobadi, Ferri), so the question is whether such rack solutions must be isomorphic. We show that they need not be. On X = Z/5 the rack solutions r(a,b) = (2b − a, a) and s(a,b) = (3a + 3b, a) of the affine quandles Aff(Z/5,2) and Aff(Z/5,3) are related by an explicit Drinfeld twist, while the two quandles are not isomorphic: the second is commutative and the first is not. Every step is an explicit finite identity, and an exact finite computation confirms all of them. More generally, Aff(Z/p,t) and its dual Aff(Z/p,t⁻¹) are non-isomorphic racks with twist-related rack solutions whenever −t has multiplicative order 3, 4 or 5 modulo the prime p; this gives such pairs of infinitely many orders. The proofs use Ferri's criterion in terms of the braid group B₃, Burnside's theorem on marks and the finite von Dyck groups. By exact computations, 5 is the smallest order of such a pair, there are 16 such pairs of order 6, and Aff(Z/7,2) is not twist-related to its dual. Status: the definite question answered negatively; the classification programme open. The classification of solutions up to Drinfeld twists stays open, and we do not address it. This is an unrefereed note.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.QA",
    "math.GR",
    "math.RA"
  ],
  "keywords": [
    "Yang-Baxter equation",
    "set-theoretic solutions",
    "Drinfeld twist",
    "racks",
    "quandles",
    "affine quandles",
    "Alexander quandles",
    "braid group B3",
    "permutation representations",
    "Burnside's theorem on marks",
    "von Dyck groups",
    "counterexample",
    "Ferri's question",
    "Oberwolfach Reports problems",
    "UnsolvedMath",
    "OWR-14299580-006",
    "math.QA",
    "math.GR",
    "math.RA",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-20",
  "publication_date": "2026-10-20",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-20",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-14299580-006/",
  "pdf_url": "https://eulersolve.org/papers/owr-14299580-006/paper.pdf?v=8b5e6e5ca268",
  "doi": "10.5281/zenodo.23509762",
  "zenodo_record_url": "https://zenodo.org/records/23509762",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "The definite question of the record is answered negatively; the programme of classifying solutions up to Drinfeld twists stays open.",
  "files": {
    "paper.pdf": {
      "sha256": "8b5e6e5ca26873767febd91fc7a1f9569906009b918177f09031e09518fa6e4b"
    },
    "source.zip": {
      "sha256": "0bb19b4d117cb280078530f51b489003af8157d5487e013825c090e2aa6bd9e8"
    },
    "verification_report.md": {
      "sha256": "61d57ac599db393d19b27e17bdcf65803f0e0298e375c98ceb61dcbc2f54521e"
    }
  },
  "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."
}