Jones Polynomial at Roots of Unity Links Knots, Quantum Groups, and TQFTs — E8 Intelligence Research

FINDING: Jones polynomial at roots of unity encodes quantum group representations, linking knot invariants to topological quantum field theories (TQFTs) and modular forms. | MATH: Jones polynomial \( V_L(t) \) specialized at \( t = e^{2\pi i / k} \) (roots of unity); quantum group \( U_q(\mathfrak{sl}_2) \) at \( q = e^{2\pi i / r} \) yields finite-dimensional representations; modular \( S \) and \( T \) matrices from conformal field theory appear as braiding and fusion matrices. | CONNECTION: Root-of-unity phases \( e^{2\pi i / k} \) generate cyclic symmetries of order \( k \); modular group \( SL(2,\mathbb{Z}) \) action on conformal blocks yields modular forms; ratios 0.618, 1.618 appear in quantum dimensions \( d_j = [2j+1]_q = \frac{q^{2j+1} - q^{-(2j+1)}}{q - q^{-1}} \) at \( q = e^{2\pi i / r} \), e.g., \( d_{1/2} = 2\cos(\pi/r) \), which for \( r=5 \) gives \( 2\cos(36^\circ) \approx 1.618 \). | DEPTH: 9 FINDING: Quantum graph algebras at roots of unity (odd order) yield new re Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com