Bridging Knot Topology and Root Systems via Alexander Invariants — E8 Intelligence Research

FINDING: Alexander polynomial invariants for knots in 3-tori and mapping tori, with connections to torus knot cone-manifold structures, provide a bridge between knot topology and lattice/root-system geometry. | MATH: Alexander polynomial Δ_K(t) for a knot K; for torus knots T(p,q), Δ(t) = (tpq−1)(t−1)/((t^p−1)(t^q−1)). Determinant |Δ(−1)| = p·q for T(p,q) — a product of coprime integers, linking to crystallographic root system ranks (e.g., Ap−1×Aq−1 structure). For knots in 3-torus T³, the polynomial generalizes via homology of the infinite cyclic cover, yielding Laurent polynomials in Z[t±1] with coefficients encoding lattice intersection forms. Cone-manifold addendum (arXiv:1101.1620) treats torus knots with cone angles 2π/n, whose holonomy lies in SL(2,C) with traces related to 2cos(π/n) — roots of unity tied to base-60 sexagesimal fractions (e.g., 1/60 = 0.01666…, 1/30, 1/20, 1/15, 1/12, 1/10, 1/6, 1/5, 1/4, 1/3, 1/2 — all with terminating sexagesimal expansions). | CON Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com