Golden Ratio Knots: Jones Polynomial at 10th Root of Unity for Fibonacci Anyons — E8 Intelligence Research

FINDING: The Jones polynomial at q = e^{iπ/5} (a 10th root of unity) is the central algebraic invariant for Fibonacci anyon braiding, linking knot theory to topological quantum computation via the golden ratio. | MATH: Jones polynomial V_L(t) satisfies skein relation t^{-1}V_{L+} − tV_{L−} = (t^{1/2} − t^{−1/2})V_{L0}. At t = e^{iπ/5}, the bracket polynomial ⟨L⟩ = (−A² − A^{−2}) with A = e^{−iπ/10} yields V_L(e^{iπ/5}) = (−A³)^{−w(L)}⟨L⟩. Fibonacci anyon fusion: τ ⊗ τ = 1 ⊕ τ, with quantum dimension d_τ = φ = (1+√5)/2 = 1.618…, and braid matrices R and F satisfying the pentagon/hexagon equations. The Jones polynomial at this root of unity evaluates to a sum over Fibonacci anyon paths: V_L(e^{iπ/5}) = Σ_{paths} d_τ^{#loops} (phase factors). | CONNECTION: **Direct golden ratio embedding.** φ = 1.618 appears as the quantum dimension of the Fibonacci anyon, and 1/φ = 0.618, φ−1 = 0.618, φ² = 2.618. The 10th root of unity e^{iπ/5} = cos(π/5) + i sin(π/5) = (φ/2) + i(√(3−φ)/2) — its real par Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com