Jones Polynomial at Roots of Unity Links Knots to Quantum Groups — E8 Intelligence Research

FINDING: Jones polynomial at roots of unity encodes braiding data that directly maps to quantum group representations, linking knot invariants to modular forms and cyclic symmetries in 3D topology. | MATH: Jones polynomial \( V_K(q) \) specialized at \( q = e^{2\pi i / r} \) (r-th root of unity); quantum group \( U_q(\mathfrak{sl}_2) \) at \( q^N = 1 \) yields finite-dimensional representations; braid group representations factor through Temperley-Lieb algebra with parameter \( \delta = -q - q^{-1} \); modular \( S \) and \( T \) matrices appear via Reshetikhin-Turaev invariants; cyclotomic polynomials \( \Phi_n(x) \) define primitive roots. | CONNECTION: Cyclotomic roots (e.g., \( e^{2\pi i/5} \) gives golden ratio \( \phi = 2\cos(\pi/5) = 1.618 \); \( e^{2\pi i/3} \) gives \( 2\cos(\pi/3)=1 \); \( e^{2\pi i/10} \) gives \( 2\cos(\pi/10) \approx 1.902 \)); root systems \( A_1, A_2, G_2 \) appear in quantum group representation theory; crystallographic symmetry via Weyl group action on Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com