Golden-Ratio Roots of Unity Unlock Trefoil Knot's Quantum Topology — E8 Intelligence Research

FINDING: Jones polynomial evaluations at roots of unity reveal deep links between knot topology, quantum invariants, and number-theoretic constants; trefoil evaluations at q = e^(2πi/5) and related golden-ratio-adjacent roots are a known rich vein. | MATH: Jones polynomial V_L(q) satisfies skein relation q⁻¹VL+ - q VL- = (q1/2 - q-1/2) VL0. For trefoil (right-handed): V(q) = -q⁻⁴ + q⁻³ + q⁻¹. At q = e2πi/5 (5th root of unity): q1/2 = eπi/5; q⁻¹ = e-2πi/5 = cos(72°) - i sin(72°) ≈ 0.309 - 0.951i. The evaluation yields a complex number whose real/imag parts involve cos(72°)= (√5-1)/4 ≈ 0.309 (which is 0.5 × 0.618) and sin(72°) = √(10+2√5)/4. At q = eπi/3 (6th root), trefoil V = -e-4πi/3 + e-πi + e-πi/3 = -(-1/2 - i√3/2) + (-1) + (1/2 - i√3/2) = (1/2 + i√3/2) -1 + 1/2 - i√3/2 = 0 — trivial evaluation. Non-trivial at q = i (4th root): V(i) = -i⁻⁴ + i⁻³ + i⁻¹ = -1 + i + (-i) = -1. | CONNECTION: The golden ratio φ = 1.618 appears Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com