Cubic Roots of Unity in SU(2)_k Knot Polynomials and Quantum A-Polynomials — 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 quantum A-polynomials and higher-rank Alexander generalizations. MATH: - Quantum group \( SU(2)_k \): deformation parameter \( q = e2π i/(k+2) \). For \( k=1 \), \( q = e2π i/3 = ω \), a primitive cubic root of unity. - Skein relation for knot polynomials at \( q=ω \): \( q1/2 - q-1/2 = ω1/2 - ω-1/2 = i√3 \) (since \( ω = -1/2 + i√3/2 \)). - Quantum A-polynomials: recurrence operators annihilating colored Jones polynomials; at \( q=ω \), reduce to classical A-polynomials with cubic cyclotomic factors. - Higher-rank generalization (CKVK*): Alexander polynomial from quantum groups of rank >1, involving roots of unity at levels corresponding to Coxeter number. CONNECTION: - Cubic roots of unity (\( ω, ω^2 \)) are intimately tied to the golden ratio via \( ω + ω^2 = Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com