Bi-orderable knot groups force positive real roots in Alexander polynomials — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
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 ⊂ Y be a knot in a rational homology 3-sphere, G = π1(Y ∖ K). If G is bi-orderable and K is rationally homologically fibered (i.e., the commutator subgroup G' is finitely generated over ℚ), then ΔK(t) has at least one root r ∈ ℝ>0. The group is an HNN extension G = H *ϕ with stable letter t, and the Alexander polynomial satisfies ΔK(t) ∼ det(t · Id − ϕ*) 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 ϕ* on H1(G';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
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BibTeX
@misc{caldin2026biorderable,
title = {{Bi-orderable knot groups force positive real roots in Alexander polynomials — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23447082},
url = {https://zenodo.org/records/23447082},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Bi-orderable knot groups force positive real roots in Alexander polynomials — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/18
PB - Zenodo
DO - 10.5281/zenodo.23447082
UR - https://zenodo.org/records/23447082
AB - 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 ⊂ Y be a knot in a rational homology 3-sphere, G = π_1(Y ∖ K). If G is bi-orderable and K is rationally homologically fibered (i.e., the commutator subgroup G' is finitely generated over ℚ), then Δ_K(t) has at least one root r ∈ ℝ_>0. The group is an HNN extension G = H *_ϕ with stable letter t, and the Alexander polynomial satisfies Δ_K(t) ∼ det(t · Id − ϕ_*) 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 ϕ_* on H_1(G';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
ER -