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

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 = \( 2 · Vol(regular ideal tetrahedron) = 2 · ( Im(Li_2(eiπ/3)) ) = 2 · 1.0149416... ≈ 2.029883... \). The regular ideal tetrahedron has dihedral angles \( π/3 \), and its volume involves the Lobachevsky function \( Λ(π/3) \). The invariant trace field is \( Q(√-3) \), and the cusp shape is \( Z[ω] \) with \( ω = eiπ/3 \). The golden ratio \( φ = (1+√5)/2 \) 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 \( φ \) 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