Alexander Polynomial's Positive Root Condition Obstructs Bi-Orderability — E8 Intelligence Research

Abstract

FINDING: Alexander polynomial of rationally homologically fibered knots in rational homology 3-spheres obstructs bi-orderability if it has no positive real root. | MATH: Let \( G = \pi_1(S^3 \setminus K) \) for a knot \( K \) in a rational homology 3-sphere. If \( G \) is bi-orderable, then the Alexander polynomial \( \Delta_K(t) \in \mathbb{Z}[t^{\pm1}] \) must have at least one positive real root. Equivalently, if \( \Delta_K(t) \) has no positive real root, \( G \) cannot be bi-orderable. The group is an HNN extension of a finitely generated group. | CONNECTION: Positive real roots of Alexander polynomials often relate to geometric ratios; e.g., roots on the unit circle correspond to complex phases, but positive real roots (e.g., \( t = \phi^2 \approx 2.618 \)) link to hyperbolic volume or growth rates. No direct golden ratio or base-60 found, but root structure may reflect lattice periodicity in the Magnus matrix. | DEPTH: 6 — Connects algebraic knot theory (Alexander polynomial) t Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com