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 \( q = e2π i / \), \(\) odd integer. The quantum group \( U_q(g) \) at root of unity yields a ribbon category with braiding \( R \)-matrix satisfying \( R^2 = qᵗ \). Invariants like the colored Jones polynomial at \( q \) are given by \( J_K(n) = Tr( ρ_n(K) ) \) where \( ρ_n \) is a highest weight representation. The modular \( S \)-matrix entries are \( Sλμ = const · ∑w ∈ W ε(w) q2(λ+ρ, w(μ+ρ)) \). | CONNECTION: Root system lattices (e.g., \( A_n, D_n, E_8 \)) determine the weight lattice and Weyl group \( W \); the ratio \( q² \) yields phases that are roots of unity, linking to crystallographic symmetry. The golden ratio appears when \( = 5 \): \( q = e2π i/5 \), and \( q + q^4 = 2cos(2π/5) = φ⁻¹ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com