Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 11, Number 3 (2011), 1323-1343.
The moduli space of hex spheres
A hex sphere is a singular Euclidean sphere with four cone points whose cone angles are (integer) multiples of but less than . 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 satisfies the following properties:
(1) it has an embedded (open Euclidean) annulus that is disjoint from the singular locus of ;
(2) it embeds isometrically in the 3–dimensional Euclidean space as the boundary of a tetrahedron; and
(3) there is a simple closed geodesic in such that a fractional Dehn twist along converts to the double of a parallelogram.
Algebr. Geom. Topol., Volume 11, Number 3 (2011), 1323-1343.
Received: 31 October 2010
Revised: 26 January 2011
Accepted: 15 February 2011
First available in Project Euclid: 19 December 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 57M50: Geometric structures on low-dimensional manifolds
Secondary: 57M15: Relations with graph theory [See also 05Cxx]
Cruz-Cota, Aldo-Hilario. The moduli space of hex spheres. Algebr. Geom. Topol. 11 (2011), no. 3, 1323--1343. doi:10.2140/agt.2011.11.1323. https://projecteuclid.org/euclid.agt/1513715229