Golden Ratio Squared Emerges as Mahler Measure of Figure-Eight Knot — E8 Intelligence Research

FINDING: The figure-eight knot's Alexander polynomial is Δ(t) = t² − 3t + 1, whose Mahler measure equals the golden ratio squared (φ² ≈ 2.618), linking knot topology to quadratic golden-ratio structure. | MATH: Δ₄₁(t) = t² − 3t + 1; roots: t = (3 ± √5)/2 = φ² and φ⁻² (where φ = (1+√5)/2 ≈ 1.618); Mahler measure M(Δ) = max(|φ²|, |φ⁻²|) = φ² ≈ 2.618; also Δ(1) = −1 (knot determinant = 1, consistent with figure-eight being amphichiral); the checkerboard matrix of the knot yields the same quadratic form. | CONNECTION: Direct — the Alexander polynomial's roots are exactly φ² and φ⁻², where φ² = 2.618 and φ⁻² = 0.382 (the complementary golden ratio). The figure-eight complement is tessellated by regular ideal hyperbolic tetrahedra (the 4₁ knot is the unique knot whose complement admits this), and its volume ≈ 2.02988... relates to the Lobachevsky function at π/3 — the same dihedral angle structure as the tetrahedral root system A₃. The Mahler measure being φ² places this knot in the class of Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com