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 \(t^{-1}V_{L_+} - tV_{L_-} = (t^{1/2} - t^{-1/2})V_{L_0}\). At \(t = e^{2\pi i/5}\), the quantum dimension \(d = \phi = (1+\sqrt{5})/2 \approx 1.618\) (Fibonacci anyon fusion rule \(\phi \otimes \phi = 1 \oplus \phi\)). R-matrix eigenvalues at 5th root: \(\lambda_1 = e^{4\pi i/5}, \lambda_2 = -e^{-2\pi i/5}\), satisfying \(\lambda_1/\lambda_2 = -\phi^2\). Fibonacci numbers appear via \(F_{n+1} = \phi F_n + (-\phi)^{-n}\) (Binet form). | CONNECTION: Golden ratio \(\phi = 1.618\) and its inverse \(\phi^{-1} = 0.618\) are the quantum dimensions of the Fibonacci anyon — the simplest non-Abelian anyon model. The 5th root of unity \(e^{2\pi i/5}\) 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