Golden Ratio Growth in Fibered Knot Alexander Eigenvalues — E8 Intelligence Research

Abstract

FINDING: Alexander polynomial eigenvalues of fibered knots exhibit growth rates linked to monodromy; golden ratio appears as a specific growth eigenvalue in certain knot families. | MATH: For a fibered knot \(K\) with monodromy \(h: F \to F\), the Alexander polynomial \(\Delta_K(t)\) satisfies \(\Delta_K(t) = \det(tI - h_*)\) where \(h_*\) acts on \(H_1(F;\mathbb{Z})\). The growth of eigenvalues of \(h_*\) (spectral radius \(\rho\)) determines the entropy of the monodromy. For the figure-eight knot (\(4_1\)), \(\Delta_K(t) = t^2 - 3t + 1\), with eigenvalues \(\phi^2 = (3+\sqrt{5})/2 \approx 2.618\) and \(\phi^{-2} \approx 0.382\), where \(\phi = (1+\sqrt{5})/2 \approx 1.618\). The ratio \(\phi^2 : 1 : \phi^{-2}\) mirrors the golden ratio squared. | CONNECTION: Eigenvalues \(\phi^2\) and \(\phi^{-2}\) are directly the squares of the golden ratio \(\phi\). The ratio 0.382 and 2.618 are the fundamental golden ratio reciprocals and squares. This links knot monodromy to the geometric harmon Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com