Cubic Roots of Unity in Quantum Knot Polynomials via SU(2)_1 Deformation — E8 Intelligence Research

FINDING: Quantum group deformation SU(2)_k at level k=1 yields cubic roots of unity in knot polynomial skein relations, linking to q-Zeilberger algorithm for quantum A-polynomials and higher-rank Alexander generalizations. MATH: - SU(2)_k: deformation parameter q = exp(2πi/(k+2)). For k=1, q = exp(2πi/3) = -1/2 + i√3/2, a primitive cubic root of unity. - Skein relation for knot polynomials at q³=1: (q - q⁻¹) factor vanishes, reducing to classical Alexander polynomial specialization. - Quantum A-polynomials: recurrence operators satisfying q-difference equations, solved via q-Zeilberger algorithm (q-hypergeometric summation). - Higher-rank generalization (CKVK*): Alexander polynomial from quantum groups of rank >1, involving root system weights and q-deformed Casimirs. CONNECTION: - Cubic roots of unity (q³=1) correspond to 120° rotations in complex plane, matching 3-fold rotational symmetry of triangular/hexagonal lattices (crystallographic point group D₃). - q = exp(2πi/ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com