q-Zeilberger Algorithm Links Knot Invariants to Recurrence at Roots of Unity — E8 Intelligence Research

FINDING: The q-Zeilberger algorithm computes quantum A-polynomials for knots, linking quantum invariants to recurrence relations at roots of unity, with skein relations encoding topological structure. MATH: Quantum A-polynomials are operator-valued polynomials in \( x̂ \) and \( ŷ \) satisfying \( Â(x̂, ŷ; q) · colored Jones polynomial = 0 \). At cubic roots of unity (\( q = e2π i/3 \)), the q-Zeilberger algorithm yields closed-form recurrences. Skein relation: \( qⁿ Jₙ₊₁ - q⁻ⁿ Jₙ₋₁ = (q1/2 - q-1/2) J_n \) for the Jones polynomial. CONNECTION: Cubic roots of unity (\( q^3 = 1 \)) correspond to 3-fold rotational symmetry, linking to crystallographic point groups (e.g., trigonal system). The golden ratio appears in quantum group \( U_q(sl_2) \) at \( q = e2π i/5 \) (not cubic), but cubic roots give \( q + q⁻¹ = -1 \), a ratio related to the silver mean (\( 1 + √2 ≈ 2.414 \)), not directly to Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com