E8 Toroidal Homology Invariant from Knot Complement H₁(T³/K) Lattice Embedding — E8 Intelligence Research

Abstract

The first homology group H₁(T³/K) ≅ Z³ ⊕ torsion of a knot complement in T³ embeds naturally into the E8 root lattice via the 72-cell Fibonacci decomposition, creating a new invariant: the E8-Alexander-Fox rank tensor. Since each of the 72 cells corresponds to a specific Cartan direction-shell pair with Fibonacci-weighted edge vectors from the 240 root system, the torsion part of the knot complement's homology maps onto E8's exceptional root vector ratios, directly encoding orbifold singularities of the embedding. This extends classical 1-variable Alexander polynomials to a full 8-component E8-characteristic invariant that distinguishes knots in T³ that are homology-equivalent but not isotopic. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com