{
  "schema_version": 1,
  "problem_number": "AMR-103-0230",
  "title": "Determinants of Prime Alternating Achiral Knots: A Partial Answer to a Problem of Stoimenow",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Stoimenow asked (Problem 12.25 of the problem list edited by Ohtsuki, 2002; Conjecture 4.2 of his paper in Comm. Anal. Geom. 13 (2005)) whether every odd sum of two squares n ∉ {1, 9, 49} is the determinant of a prime alternating achiral knot. He proved this for squares and, with rational knots, for all n > 1 without prime factors q ≡ 3 (mod 4), and reported a check up to 2000. We give a partial answer; the question remains open. Gluing a plane network in a disk to its dual network, turned over by a half-turn of the sphere, is a form of a classical construction of self-dual maps; when the resulting graph is nonseparable and has an odd number of spanning trees, it is the checkerboard graph of a prime alternating achiral knot. For two and three boundary points this gives Stoimenow's determinant formulas back; for four it gives a new one. With it we show: (i) if n is the determinant of such a knot, so are s⁴n for odd s ≥ 3 and 3⁶n; hence so is every n = j⁴m with j odd and m > 1 composed of primes ≡ 1 (mod 4), and the question reduces to n = d²m with d > 1 a squarefree product of primes ≡ 3 (mod 4); (ii) 5q² is the determinant of such a knot for every prime q ≡ 3 (mod 4), which extends Stoimenow's remark on q ≤ 11; (iii) so is every number in an explicit range of the form d²(A² + B²), by three-terminal graphs; (iv) so is every n ≤ 10⁶ not covered by Stoimenow's results (16,703 numbers), by explicit certificates checked by computer. For larger n the general case is open, and we explain why these families do not settle it. This is an unrefereed note.",
  "result_type": "COMPLETE_SCOPED_PROOF",
  "categories": [
    "math.GT",
    "math.CO"
  ],
  "keywords": [
    "achiral knots",
    "amphicheiral knots",
    "alternating knots",
    "prime knots",
    "knot determinant",
    "sums of two squares",
    "self-dual plane graphs",
    "spanning trees",
    "checkerboard graph",
    "Stoimenow conjecture",
    "partial answer",
    "Ohtsuki problem list",
    "UnsolvedMath",
    "AMR-103-0230",
    "math.GT",
    "math.CO",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-19",
  "publication_date": "2026-10-19",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-19",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-103-0230/",
  "pdf_url": "https://eulersolve.org/papers/amr-103-0230/paper.pdf?v=5e022ae3e3c2",
  "doi": "10.5281/zenodo.23482759",
  "zenodo_record_url": "https://zenodo.org/records/23482759",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Partial answer to Stoimenow's Problem 12.25: the question stays open for general n = d^2 m above 10^6; Stoimenow's results, formulas (9) and (12) and his remark on 8* are credited.",
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    "verification_report.md": {
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  "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."
}