A Structural Reclassification of Thurston's Eight Geometries Through Knot Complements

Abstract

This paper reinterprets Thurston’s eight geometries through knot complements, showing how reflection, closure, and chirality encode geometric phases. The framework reveals non‑commutative residues distinguishing Nil, Sol, and SL2R~ from commutative geometries. This work is licensed under the Creative Commons Attribution 4.0 International (CC BY 4.0) license.