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 \( t = e2π i / k \) (roots of unity); quantum group \( U_q(sl_2) \) at \( q = e2π i / r \) yields finite-dimensional representations; modular \( S \) and \( T \) matrices from conformal field theory appear as braiding and fusion matrices. | CONNECTION: Root-of-unity phases \( e2π i / k \) generate cyclic symmetries of order \( k \); modular group \( SL(2,Z) \) action on conformal blocks yields modular forms; ratios 0.618, 1.618 appear in quantum dimensions \( d_j = [2j+1]_q = {q²ʲ⁺¹ - q⁻⁽²ʲ⁺¹⁾}{q - q⁻¹} \) at \( q = e2π i / r \), e.g., \( d1/2 = 2cos(π/r) \), which for \( r=5 \) gives \( 2cos(36^∘) ≈ 1.618 \). | 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