Alexander Polynomials for Knots in the 3-Torus — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
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
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Related
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- The Alexander polynomial of twisted torus knots in the complementary range
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- Multivariable Alexander polynomials of twisted torus links via Dehn surgery
BibTeX
@misc{caldin2026alexander,
title = {{Alexander Polynomials for Knots in the 3-Torus — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23431164},
url = {https://zenodo.org/records/23431164},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Alexander Polynomials for Knots in the 3-Torus — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/17
PB - Zenodo
DO - 10.5281/zenodo.23431164
UR - https://zenodo.org/records/23431164
AB - 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 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -