Alexander Polynomials for Knots in the 3-Torus — E8 Intelligence Research

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