On irreducible triangulations of punctured and pinched surfaces
M. J. Ch\'avez (1), S. Lawrencenko (2), A. Quintero (3), M. T. Villar, (3) ((1) Departamento de Matem\'atica Aplicada I, Universidad de Sevilla,, Sevilla, Spain, (2) Department of Higher Mathematics 1, National Research, University of Electronic Technology, Zelenograd, Russia
TL;DR
This paper proves the finiteness of irreducible triangulations for punctured and pinched surfaces, providing complete classifications for specific cases like the Möbius band, pinched torus, and projective plane with up to 8 vertices.
Contribution
It establishes the finiteness of irreducible triangulations for punctured surfaces and provides explicit classifications for several key surface types.
Findings
Finiteness of irreducible triangulations for punctured surfaces.
Complete lists of irreducible triangulations for the Möbius band and pinched torus.
Classification of triangulations of the projective plane with up to 8 vertices.
Abstract
A triangulation of a punctured or pinched surface is irreducible if no edge can be shrunk without producing multiple edges or changing the topological type of the surface. The finiteness of the set of (non-isomorphic) irreducible triangulations of any punctured surface is established. Complete lists of irreducible triangulations are determined for the M\"obius band (6 in number) and the pinched torus (2 in number). All the non-isomorphic combinatorial types (20 in number) of triangulations of the projective plane with up to 8 vertices are determined.
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Taxonomy
TopicsComputational Geometry and Mesh Generation · Digital Image Processing Techniques · Topological and Geometric Data Analysis