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arXiv · 1711.07603

Non-spanning lattice 3-polytopes

Mónica Blanco·Francisco Santos
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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: $P\cap \mathbb{Z}^3$ 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.

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Mónica Blanco, Francisco Santos. 2018-10-01. Non-spanning lattice 3-polytopes. https://doi.org/10.1016/j.jcta.2018.07.010

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