Geometrical Spines of Lens Manifolds

We introduce the concept of ‘geometrical spine’ for 3‐manifolds with natural metrics, in particular, for lens manifolds. We show that any spine of L p,q that is close enough to its geometrical spine contains at least E ( p,q ) − 3 vertices, which is exactly the conjectured value for the complexity c ( L p,q ). As a byproduct, we find the minimal rotation distance (in the Sleator–Tarjan–Thurston sense) between a triangulation of a regular p ‐gon and its image under rotation.