{
  "schema_version": 1,
  "problem_number": "AMR-103-0226",
  "title": "Knots with Signature 4 and Determinant 4k+1: An Answer to a Question of Shinohara",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "By a congruence of Murasugi, the determinant and the signature of a knot K satisfy det K ≡ (−1)^(σ(K)/2) (mod 4); moreover, a knot of determinant 1 has signature divisible by 8, because its symmetrized Seifert form is even and unimodular. In 1971 Shinohara realized all pairs (det K, |σ(K)|) allowed by these conditions, except those with det K ≡ 1 (mod 8) and |σ(K)| ≡ 4 (mod 8); he stated that the case det K ≢ 1 (mod 24) can be treated as well and left the case det K ≡ 1 (mod 24) open, expecting an affirmative answer. The question reappeared as Problem 12.21 of Ohtsuki's list of problems on invariants of knots and 3-manifolds: is there, for every n = 4k + 1 with k > 0, a knot with determinant n and signature 4? Stoimenow showed that every prime factor of a counterexample is ≡ 1 (mod 24) and at least 33049, and asked whether {det K : σ(K) = 4} = 5 + 4ℕ. We answer these questions affirmatively. For every n ≡ 1 (mod 4) with n ≥ 5 there is a knot of determinant n and signature 4; it can be taken of genus 2 unless n is a square all of whose prime factors are ≡ 1 (mod 4), and of genus at most 3 in that case. Consequently, an odd positive integer d and an even integer s are the determinant and the signature of some knot if and only if d ≡ (−1)^(s/2) (mod 4), where d = 1 requires s ≡ 0 (mod 8). The proof combines Seifert's realization theorem with an explicit family of genus-two Seifert matrices and a lemma based on Dirichlet's theorem on primes in arithmetic progressions and quadratic reciprocity. We also give explicit knots, among them a two-bridge knot of determinant 33049, and report exact computer checks. This is an unrefereed note.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.GT",
    "math.NT"
  ],
  "keywords": [
    "knot theory",
    "knot signature",
    "knot determinant",
    "Seifert matrix",
    "Murasugi congruence",
    "Dirichlet's theorem",
    "quadratic reciprocity",
    "even lattices",
    "Ohtsuki problem list",
    "Problem 12.21",
    "Shinohara",
    "UnsolvedMath",
    "AMR-103-0226",
    "math.GT",
    "math.NT",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-02",
  "publication_date": "2026-10-02",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-02",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-103-0226/",
  "pdf_url": "https://eulersolve.org/papers/amr-103-0226/paper.pdf?v=8db411591685",
  "doi": "10.5281/zenodo.23111900",
  "zenodo_record_url": "https://zenodo.org/records/23111900",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Answers Problem 12.21 of Ohtsuki's list (Geom. Topol. Monogr. 4, 2002; due to Y. Shinohara) and Stoimenow's Question 5.1 (Acta Arith. 2007) affirmatively: every n = 4k+1 >= 5 is the determinant of a knot of signature 4, which gives the complete list of (determinant, signature) pairs of knots. Shinohara (1971) and Stoimenow (2007) had settled all cases except determinants whose prime factors are all 1 mod 24; the new part is an explicit genus-2 family of Seifert matrices with a lemma based on Dirichlet's theorem and quadratic reciprocity. A lattice-theoretic route may make the result known to experts. Whether genus 2 suffices for every square determinant with all prime factors 1 mod 4 remains open.",
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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."
}