Bi-orderable knot groups force positive real roots in Alexander polynomials — E8 Intelligence Research

FINDING: Rationally homologically fibered knot groups with bi-orderable fundamental groups force the Alexander polynomial to have a positive real root — a topological obstruction linking orderability to spectral properties. | MATH: Let \(K \subset Y\) be a knot in a rational homology 3-sphere, \(G = \pi_1(Y \setminus K)\). If \(G\) is bi-orderable and \(K\) is rationally homologically fibered (i.e., the commutator subgroup \(G'\) is finitely generated over \(\mathbb{Q}\)), then \(\Delta_K(t)\) has at least one root \(r \in \mathbb{R}_{>0}\). The group is an HNN extension \(G = H *_{\phi}\) with stable letter \(t\), and the Alexander polynomial satisfies \(\Delta_K(t) \sim \det(t \cdot \mathrm{Id} - \phi_*)\) up to units. The proof uses the fact that a bi-orderable group admits a Conradian left-order, and the HNN structure forces a real eigenvalue of the monodromy \(\phi_*\) on \(H_1(G';\mathbb{Q})\). | CONNECTION: The positive real root condition is a spectral signature. For a fibered Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com