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 \( U_q(\mathfrak{g}) \) specialized at \( q = e^{2\pi i / \ell} \) (root of unity); representation theory truncates to finite set; knot invariants (e.g., colored Jones polynomial) become periodic in color with period \(\ell\); lattice structures arise from weight lattice modulo \(\ell\) times root lattice: \( P / \ell Q \). | CONNECTION: Root system lattices (e.g., \( A_n, D_n, E_8 \)) produce crystallographic symmetries; ratios like \( q^{1/2} \) phases yield cyclic groups of order \(\ell\); base-60 not directly present, but modular data from \( SL(2,\mathbb{Z}) \) action on characters involves \( e^{2\pi i k / \ell} \) 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