Figure-Eight Knot Complement: Golden Ratio, Ideal Tetrahedron, and Hyperbolic Volume — E8 Intelligence Research

Abstract

FINDING: Figure-eight knot complement is the simplest hyperbolic knot complement, with volume linked to the golden ratio via ideal tetrahedron decomposition and Dehn surgery limits. MATH: Volume of figure-eight knot complement = \( 2 \cdot \text{Vol}(\text{regular ideal tetrahedron}) = 2 \cdot ( \text{Im}(\text{Li}_2(e^{i\pi/3})) ) = 2 \cdot 1.0149416... \approx 2.029883... \). The regular ideal tetrahedron has dihedral angles \( \pi/3 \), and its volume involves the Lobachevsky function \( \Lambda(\pi/3) \). The invariant trace field is \( \mathbb{Q}(\sqrt{-3}) \), and the cusp shape is \( \mathbb{Z}[\omega] \) with \( \omega = e^{i\pi/3} \). The golden ratio \( \phi = (1+\sqrt{5})/2 \) appears in the complex length of the meridian under certain Dehn fillings; the figure-eight knot is the \( (2, -3) \) twist knot, and its hyperbolic structure relates to the arithmetic of \( \phi \) via the polynomial \( x^2 - x - 1 = 0 \) appearing in the holonomy. CONNECTION: The regular ideal te Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com