Quantum Group Knot Invariants at Roots of Unity and Modular Data — E8 Intelligence Research

FINDING: Quantum group knot invariants at roots of unity produce finite-dimensional braided tensor categories with modular data linked to root system lattices. | MATH: Let \( q = e^{2\pi i / \ell} \), \(\ell\) odd integer. The quantum group \( U_q(\mathfrak{g}) \) at root of unity yields a ribbon category with braiding \( R \)-matrix satisfying \( R^2 = q^{t} \). Invariants like the colored Jones polynomial at \( q \) are given by \( J_K(n) = \text{Tr}( \rho_n(K) ) \) where \( \rho_n \) is a highest weight representation. The modular \( S \)-matrix entries are \( S_{\lambda\mu} = \text{const} \cdot \sum_{w \in W} \epsilon(w) q^{2(\lambda+\rho, w(\mu+\rho))} \). | CONNECTION: Root system lattices (e.g., \( A_n, D_n, E_8 \)) determine the weight lattice and Weyl group \( W \); the ratio \( q^{2} \) yields phases that are roots of unity, linking to crystallographic symmetry. The golden ratio appears when \( \ell = 5 \): \( q = e^{2\pi i/5} \), and \( q + q^4 = 2\cos(2\pi/5) = \phi^{-1} \a Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com