q-Zeilberger Algorithm Links Knot Skein Relations to Hypergeometric Structures — E8 Intelligence Research

FINDING: The q-Zeilberger algorithm applied to quantum A-polynomials at cubic roots of unity generates knot polynomial skein relations, linking quantum invariants to recursive hypergeometric structures. MATH: - Quantum A-polynomial: \( Â(x, y; q) Ψ(x) = 0 \), where \( y Ψ(x) = Ψ(qx) \), \( x Ψ(x) = x Ψ(x) \). - At cubic roots of unity: \( q = e2π i / 3 \), so \( q^3 = 1 \), \( 1 + q + q^2 = 0 \). - Skein relation for Jones polynomial: \( q^2 V(L_+) - q⁻² V(L_-) = (q - q⁻¹) V(L_0) \). - q-Zeilberger algorithm reduces recurrence order via telescoping sums, yielding closed-form \( q \)-hypergeometric series for colored Jones polynomials. CONNECTION: - Cubic root of unity (\( q = e2π i/3 \)) corresponds to 3-fold rotational symmetry, linking to crystallographic point group \( C_3 \) and root system \( A_2 \). - The ratio \( q + q⁻¹ = -1 \) appears, which is related to the golden ratio via \( φ = 2cos(π/5) \), but here \( 2cos(2 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com