Golden Ratio Links Hyperbolic Volume to Colored Jones Polynomial Asymptotics — E8 Intelligence Research

FINDING: The colored Jones polynomial's asymptotics connect hyperbolic volume to the golden ratio via the volume conjecture, with the figure-eight knot as the canonical test case. | MATH: Volume conjecture: \( \lim_{N \to \infty} \frac{2\pi \log |J_N(K; e^{2\pi i/N})|}{N} = \text{Vol}(S^3 \setminus K) \). For figure-eight knot \(4_1\), \( \text{Vol} = 2V_3 = 2 \times 1.01494... = 2.02988... \), where \(V_3\) is the Lobachevsky function at \(\pi/3\). The Mahler measure of the colored Jones polynomial for \(4_1\) on the unit circle yields \(m(J_N) \to \text{Vol}/2\pi\). The golden ratio appears in the figure-eight knot's hyperbolic structure: the cusp shape is \( \pm 2 + \sqrt{3}i \) (not golden), but the trace field is \(\mathbb{Q}(\sqrt{-3})\), and the volume is related to the dilogarithm at \(e^{i\pi/3}\). | CONNECTION: The golden ratio \(\phi = 1.618\) enters via the *geometric* realization of the figure-eight knot complement as a regular ideal octahedron with dihedral angles \(\pi/3 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com