{
  "schema_version": 1,
  "problem_number": "AIM-TOPOLOGY-0043",
  "title": "Quantum Integer Torsion in the Skein Module of a Connected Sum of Two Solid Tori",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We extract an explicit infinite family of torsion elements from the two-solid-torus presentation of Bakshi, Le and Przytycki. Over R=Z[q,q^{-1}], put B=R[x_1,x_2] and [m]=(q^{2m}-q^{-2m})/(q^2-q^{-2}). For every m>=2 we construct an element whose exact B-annihilator is ([m]), and these elements embed the direct sum of B/([m]) into the torsion submodule. Thus no single nonzero Laurent polynomial annihilates all torsion. A further explicit element has annihilator (q^8-q^4+1), comaximal with (q^2-q^{-2}). The calculation answers the coprime-torsion question in arXiv:2604.09971v1 and contradicts its bounded-annihilator corollary; the gap is an identification of an unsaturated quotient with its fraction-field image. Our argument is algebraic and uses the cited presentation for its skein interpretation. It does not compute the skein module of arbitrary closed connected sums of lens spaces. This is a self-audited, unrefereed, AI-assisted preprint. No independent review, formal verification or absolute priority is claimed. The source package contains the complete proof, detailed audit and exact-arithmetic checker with 156 passing regression assertions.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.GT",
    "math.QA"
  ],
  "keywords": [
    "Kauffman bracket skein module",
    "connected sum",
    "solid torus",
    "torsion",
    "quantum integer",
    "annihilator",
    "saturation",
    "AIM-TOPOLOGY-0043"
  ],
  "manuscript_version_date": "2026-10-06",
  "publication_date": "2026-10-06",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-06",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-topology-0043/",
  "pdf_url": "https://eulersolve.org/papers/aim-topology-0043/paper.pdf?v=4846a46f611e",
  "doi": "10.5281/zenodo.23172763",
  "zenodo_record_url": "https://zenodo.org/records/23172763",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete algebraic theorems for the two-solid-torus presentation of Bakshi–Lê–Przytycki (arXiv:2604.09971v1, Theorem 3.1), including exact quantum-integer annihilators and an infinite direct-sum torsion submodule. The skein interpretation uses that cited presentation. The result answers Question 1 and corrects the bounded-annihilator assertion in Corollary 3.12(a); it does not refute the main presentation or establish that the splitting assertion is false. Arbitrary closed connected sums of lens spaces, a full torsion classification and survival under filling are not settled. The 156 exact regression assertions supplement the written proof. AI-assisted, self-audited and unrefereed; independent review, formal verification and absolute priority are not claimed. Novelty remains undetermined after a bounded primary-source search.",
  "files": {
    "paper.pdf": {
      "sha256": "4846a46f611e56cfa058e9288badb1f125d3aa74983e8d8a39aae8f94465710f"
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    "source.zip": {
      "sha256": "9c5a98376c1984a012a0e99d97d4a057c4fee66f6d22e26a4296a80d50ac7d55"
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    "verification_report.md": {
      "sha256": "04d02cb1e38bae14e0e95373e9419ca66abddcafc3f82ba73dde30daad49cdf2"
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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."
}