2008/09/11 by Taiyo Inoue
Computer Science · Mathematics · #Class (philosophy) #Combinatorics #Computational Geometry and Mesh Generation #Computer science #Dimension (graph theory) #Disjoint sets #Disjoint union (topology) #Dodecahedron #Geometric and Algebraic Topology #Geometry #Mathematics #Polyhedron #math.GT #semigroups and automata theory
paper · pdf · doi:10.2140/agt.2008.8.1523
published as Algebr. Geom. Topol. 8 (2008) 1523-1565 · 34 pages, 8 figures, to appear in Algebraic & Geometric Topology
arxiv created 2008/09/11 · openalex publication_date 2008/09/15 · arxiv updated 2014/09/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
This article defines a pair of combinatorial operations on the combinatorial structure of compact right-angled hyperbolic polyhedra in dimension three called decomposition and edge surgery. It is shown that these operations simplify the combinatorics of such a polyhedron, while keeping it within the class of right-angled objects, until it is a disjoint union of Lbell polyhedra, a class of polyhedra which generalizes the dodecahedron. Furthermore, these combinatorial operations are shown to have geometric realizations which are volume decreasing. This allows for an organization of the volumes of right-angled hyperbolic polyhedra and allows, in particular, the determination of the polyhedra with smallest and second smallest volumes. 51M10, 57M50; 52B99