Trefoil Jones Polynomial at 6th Root of Unity Encodes Golden Ratio — E8 Intelligence Research

FINDING: The Jones polynomial V(t) of the trefoil knot, evaluated at the primitive 6th root of unity t = e^{iπ/3}, yields a value whose modulus relates to the golden ratio φ = (1+√5)/2; the trefoil is the simplest nontrivial knot and its quantum invariant encodes SL(2,ℂ) and SU(2) representation data tied to the golden ratio at level k=3 Chern-Simons theory. MATH: - Jones polynomial of trefoil (left-handed): V(t) = −t^{−4} + t^{−3} + t^{−1} - Right-handed trefoil: V(t) = −t^4 + t^3 + t - Skein relation: t^{−1}V(L₊) − tV(L₋) = (t^{1/2} − t^{−1/2})V(L₀) - At t = e^{iπ/3} (primitive 6th root of unity, q = e^{2πi/6}): - t^3 = e^{iπ} = −1, t^4 = e^{i4π/3} = −1/2 − i√3/2 - V(trefoil, t=e^{iπ/3}) = −e^{−i4π/3} + e^{−iπ} + e^{−iπ/3} - = (1/2 + i√3/2) + (−1) + (1/2 − i√3/2) = 0 - |V| = 0 at this root — a degeneration point - At t = e^{2πi/5} (primitive 5th root, φ-related): the quantum dimension [2]_q = q + q^{−1} = 2cos(2π/5) = (√5−1)/2 = 1/φ ≈ 0.618 - Quantum invariant at level k: q Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com