Golden Ratio in Jones Polynomial: Icosahedral Symmetry via Temperley–Lieb Algebra — E8 Intelligence Research

FINDING: Jones polynomial is a knot invariant with deep ties to quantum computing and TQFT; the golden ratio appears as a special evaluation point where the polynomial relates to the icosahedral symmetry via the Fibonacci/Lucas structure of the Temperley–Lieb algebra. | MATH: Jones polynomial \(V_L(t)\) satisfies skein relation \(t⁻¹VL_+ - t VL_- = (t1/2 - t-1/2)VL_0\). At \(t = e2π i/5\) (or \(t = eiπ/5\)), the quantum invariant reduces to a value involving \(φ = (1+√5)/2\). Specifically, the Kauffman bracket at \(A = eiπ/5\) gives \(V_L(e2π i/5)\) with coefficients in \(Z[φ]\). The Temperley–Lieb algebra at this root of unity has dimension related to Fibonacci numbers \(F_n\), and the Jones–Wenzl idempotent projects onto the golden-ratio sector. The quantum group \(SU(2)_5\) (level 5) has fusion rules governed by \(φ\) — the quantum dimension of the fundamental rep is \([2]_q = q + q⁻¹ = 2cos(π/5) = φ\). | CONNECT Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com