{
  "abstract": "FINDING: Alexander polynomial theory extends from classical knots in S³ to knots embedded in the 3-torus T³, requiring a Laurent polynomial in three variables (one per torus direction), with the classical case recovered as a specialization. | MATH: Classical Alexander polynomial Δ_K(t) ∈ ℤ[t^{±1}] satisfies Δ_K(1)=±1 and Δ_K(t)=Δ_K(t^{-1}) up to units. For knots in T³, the invariant becomes Δ_K(x,y,z) ∈ ℤ[x^{±1}, y^{±1}, z^{±1}], with the 3-torus fundamental group π₁(T³) ≅ ℤ³ acting as the abelianization target. The twisted version (arXiv:2605.13288) introduces mod p reduction: Δ_{K,ρ}(t) mod p, where ρ: π₁(S³\\K) → GL(n,ℤ/pℤ) is a finite-group representation, with regular representation yielding a product formula over irreducible characters. | CONNECTION: The 3-torus T³ = S¹×S¹×S¹ is the maximal torus of the Lie group SU(4) and relates to the A₃ root system (rank 3, Weyl group S₄, order 24). The three variables correspond to the three simple roots of A₃; the lattice ℤ³ is the A₃ weight\n\nAuthor: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com",
  "abstract_shown": true,
  "affiliations": [
    "Independent Researcher, United Kingdom"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Andrew Stewart Caldin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2zfqfmga5zkqt.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23431164",
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  "id": "2zfqfmga5zkqt",
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  "landing_url": "https://zenodo.org/records/23431164",
  "language": "eng",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
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  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
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  "published": "2026-10-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2zfqfmga5zkqt.ris",
  "sources": [
    {
      "first_seen": "2026-10-31T12:22:03Z",
      "native_id": "23431164",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23431164"
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  "subjects": [],
  "title": "Alexander Polynomials for Knots in the 3-Torus — E8 Intelligence Research",
  "title_text": "Alexander Polynomials for Knots in the 3-Torus — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-10-31T12:22:03Z",
  "url": "https://research.draconitic.com/paper/2zfqfmga5zkqt",
  "venue": "Zenodo",
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