Jones Polynomial at 5th Roots Links Knots to Fibonacci Anyons — E8 Intelligence Research

FINDING: Jones polynomial at 5th roots of unity connects to Fibonacci anyons and golden-ratio braiding, linking knot invariants to topological quantum computing. | MATH: Jones polynomial \(V_L(t)\) satisfies skein relation \(t⁻¹VL_+ - tVL_- = (t1/2 - t-1/2)VL_0\). At \(t = e2π i/5\), the quantum dimension \(d = φ = (1+√5)/2 ≈ 1.618\) (Fibonacci anyon fusion rule \(φ ⊗ φ = 1 ⊕ φ\)). R-matrix eigenvalues at 5th root: \(λ_1 = e4π i/5, λ_2 = -e-2π i/5\), satisfying \(λ_1/λ_2 = -φ^2\). Fibonacci numbers appear via \(Fₙ₊₁ = φ F_n + (-φ)⁻ⁿ\) (Binet form). | CONNECTION: Golden ratio \(φ = 1.618\) and its inverse \(φ⁻¹ = 0.618\) are the quantum dimensions of the Fibonacci anyon — the simplest non-Abelian anyon model. The 5th root of unity \(e2π i/5\) generates the dihedral symmetry \(D_5\) (pentagonal, crystallographic point group 5/m in 2D), and the Jones polynomial at this Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com