The moduli space of hex spheres

Details The moduli space of hex spheres The moduli space of hex spheres

2010-10-25

arxiv.org/abs/1010.5235v1

Mathematics

Geometric Topology

Scientific paper

A hex sphere is a singular Euclidean sphere with four cone points whose cone angles are (integer) multiples of $\frac{2\pi}{3}$ but less than $2\pi$. We prove that the Moduli space of hex spheres of unit area is homeomorphic to the the space of similarity classes of Voronoi polygons in the Euclidean plane. This result gives us as a corollary that each unit-area hex sphere $M$ satisfies the following properties: (1) it has an embedded (open Euclidean) annulus that is disjoint from the singular locus of $M$; (2) it embeds isometrically in the 3-dimensional Euclidean space as the boundary of a tetrahedron; and (3) there is a simple closed geodesic $\gamma$ in $M$ such that a fractional Dehn twist along $\gamma$ converts $M$ to the double of a parallelogram.

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