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

FINDING: Jones polynomial at roots of unity encodes quantum group representations, linking knot invariants to modular forms and 3D topological quantum field theories (TQFTs). | MATH: Jones polynomial \( V_K(q) \) specialized at \( q = e^{2\pi i / r} \) (roots of unity); quantum group \( U_q(\mathfrak{sl}_2) \) at roots of unity yields finite-dimensional representations with braiding; modular \( S \) and \( T \) matrices arise from \( \text{SL}(2,\mathbb{Z}) \) action on conformal blocks; graph algebras \( \mathcal{L}_{g,n}^\varepsilon \) at odd-order roots of unity relate to surface mapping class groups. | CONNECTION: Root-of-unity phases \( e^{2\pi i / r} \) produce cyclic symmetries of order \( r \); modular forms at level \( r \) involve Dedekind eta function \( \eta(\tau) \) and Eisenstein series; root systems \( A_1, A_2, G_2 \) appear in quantum group representation theory; lattice structures from \( \text{SL}(2,\mathbb{Z}) \) modular group; no direct golden ratio or base-60 link Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com