Quantum Group Knot Invariants at Roots of Unity and Root Lattice TQFTs — 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 becomes semisimple only for certain \( \ell \); knot invariants (e.g., colored Jones polynomial) evaluated at roots of unity produce topological quantum field theory (TQFT) invariants; root system lattice \( \Lambda_R \) and Weyl group symmetry govern fusion rules. | CONNECTION: Root systems (e.g., \( A_n, D_n, E_6, E_7, E_8 \)) define lattice structures; at roots of unity, quantum group dimensions reduce to quantum integers \([n]_q = (q^n - q^{-n})/(q - q^{-1})\), which at \( q = e^{2\pi i / \ell} \) yield periodic patterns; ratios like \( \sin(\pi a/\ell)/\sin(\pi b/\ell) \) appear, linking to golden ratio when \( \ell = 5 \) (e.g., \( \sin(2\pi/5)/\sin(\pi/5) = \phi \approx 1.618 \)); base-60 emerges from \( \el Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com