Jones Polynomial at Roots of Unity Links Knots to 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 in 3D topology. | MATH: Jones polynomial \( V_L(q) \) specialized at \( q = e^{2\pi i / k} \) (roots of unity); quantum group \( U_q(\mathfrak{sl}_2) \) at \( q \) a root of unity yields finite-dimensional representations with braiding; modular \( S \) and \( T \) matrices from conformal field theory appear via Reshetikhin–Turaev invariants; graph algebras \( \mathcal{L}_{g,n}^\varepsilon \) at odd-order roots of unity \( \varepsilon \) produce 3-manifold invariants. | CONNECTION: Root-of-unity phases \( e^{2\pi i / k} \) correspond to cyclic symmetries of order \( k \); modular forms link to elliptic curves and lattice theta functions; braid group representations at roots of unity yield quantum dimensions \( d_i = \frac{\sin(\pi (i+1)/k)}{\sin(\pi/k)} \), which for \( k=5 \) give ratios \( \phi \) (golden ratio 1.618 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com