Geometrical spines of lens manifolds

Details Geometrical spines of lens manifolds Geometrical spines of lens manifolds

2005-02-16

arxiv.org/abs/math/0502326v1

Mathematics

Geometric Topology

16 pages, 6 figures, AMS-TeX

Scientific paper

Geometrical spines are defined for 3-manifolds with natural metrics, in particular, for lens manifolds. We show that any spine of L(p,q) close enough to its geometrical spine (i.e., to the cut locus with respect to the standard metric) contains at least E(p,q)-3 vertices, which is exactly the conjectured value for Matveev's complexity of L(p,q); here E(p,q) stands for the sum of the elements of the continued fraction expansion of p/q. As a byproduct, we find the minimal (over all triangulations) rotation distance (the term coined by Sleator, Tarjan, and Thurston) between a triangulation of a regular p-gon and its image under (2Pi q/p)-rotation. This minimum is also equal to E(p,q)-3.

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