Golden Ratio Growth in Alexander Eigenvalues Links Knot Monodromy Entropy — E8 Intelligence Research

Abstract

FINDING: Alexander polynomial eigenvalues for fibered knots can exhibit growth rates linked to the golden ratio, affecting monodromy entropy. MATH: Alexander polynomial Δ_K(t) eigenvalues λ_i; monodromy entropy h = Σ log|λ_i| for λ_i outside unit circle. For fibered knots with pseudo-Anosov monodromy, growth rates often involve quadratic irrationals; golden ratio φ = (1+√5)/2 ≈ 1.618 appears as eigenvalue magnitude in specific knots (e.g., certain pretzel or hyperbolic knots). CONNECTION: Golden ratio φ and its reciprocal φ⁻¹ ≈ 0.618 are eigenvalues of the monodromy matrix, reflecting hyperbolic geometry and Teichmüller theory. The ratio 0.618 appears in the trace of the monodromy (e.g., trace = φ + φ⁻¹ = √5 ≈ 2.236). This links to base-60? No direct evidence. Crystallographic symmetry? Not directly; but pseudo-Anosov maps have dilation factors that are Perron numbers, often quadratic. DEPTH: 7 — Connects knot theory, hyperbolic geometry, and dynamical entropy via algebraic invar Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com