Non-spanning lattice 3-polytopes

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:161

Article

Blanco, Monica¹  Santos, Francisco² 

[1] Math Sci Res Inst, 17 Gauss Way, Berkeley, CA 94720 USA

[2] Univ Cantabria, Dept Matemat Estadist & Comp, Av Castros 48, E-39005 Santander, Spain

关键词: Lattice polytope;    Spanning;    Classification;    3-dimensional;    Lattice width;   

DOI  :  10.1016/j.jcta.2018.07.010

来源: Elsevier

【 摘 要 】

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: P boolean AND Z(3) 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. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free