Quantum Group Knot Invariants at Roots of Unity: Finite Representations and Lattice Periodicity — E8 Intelligence Research

FINDING: Quantum group knot invariants at roots of unity yield finite-dimensional representations and link invariants tied to root system lattices. | MATH: Quantum group \( U_q(g) \) specialized at \( q = e2π i / \) (root of unity); representation theory truncates to finite set; knot invariants (e.g., colored Jones polynomial) become periodic in color with period \(\); lattice structures arise from weight lattice modulo \(\) times root lattice: \( P / Q \). | CONNECTION: Root system lattices (e.g., \( A_n, D_n, E_8 \)) produce crystallographic symmetries; ratios like \( q1/2 \) phases yield cyclic groups of order \(\); base-60 not directly present, but modular data from \( SL(2,Z) \) action on characters involves \( e2π i k / \) phases. | DEPTH: 8 — Links quantum topology to finite group theory and lattice geometry; foundational for 3-manifold invariants (Reshetikhin–Turaev, Turaev–Viro). Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com