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Non-spanning lattice 3-polytopes

Abstract: We completely classify non-spanning 3-polytopes, by which we mean lattice 3-polytopes whose lattice points do not affinely span the lattice. We show that, except for six small polytopes (all having between five and eight lattice points), every non-spanning 3-polytope P has the following simple description: consists of either (1) two lattice segments lying in parallel and consecutive lattice planes or (2) a lattice segment together with three or four extra lattice points placed in a very specific manner. From this description we conclude that all the empty tetrahedra in a non-spanning 3-polytope P have the same volume and they form a triangulation of P, and we compute the h*-vectors of all non-spanning 3-polytopes. We also show that all spanning 3-polytopes contain a unimodular tetrahedron, except for two particular 3-polytopes with five lattice points.

 Fuente: Journal of combinatorial theory. Series A, 2019, 161, 112-133

 Editorial: Elsevier

 Fecha de publicación: 01/01/2019

 Nº de páginas: 22

 Tipo de publicación: Artículo de Revista

 DOI: 10.1016/j.jcta.2018.07.010

 ISSN: 0097-3165,1096-0899

 Proyecto español: MTM2014-54207-P

 Url de la publicación: https://doi.org/10.1016/j.jcta.2018.07.010