Lattice 3-polytopes with few lattice points

Abstract

We extend White's classification of empty tetrahedra to the complete classification of lattice $3$-polytopes with five lattice points, showing that, apart from infinitely many of width one, there are exactly nine equivalence classes of them with width two and none of larger width. We also prove that, for each $n\in \mathbb{N}$, there is only a finite number of (classes of) lattice $3$-polytopes with $n$ lattice points and of width larger than one. This implies that extending the present classification to larger sizes makes sense, which is the topic of subsequent papers of ours.

19 pages, 9 figures

References in corpus (2)

• Lattice 3-polytopes with six lattice points
• Clean Lattice Tetrahedra

Cited by in corpus (9)

• Lattice 3-polytopes with six lattice points
• Enumeration of lattice 3-polytopes by their number of lattice points
• The complete classification of empty lattice $4$-simplices
• Classification of triples of lattice polytopes with a given mixed volume
• Non-spanning lattice 3-polytopes
• The Finiteness Threshold Width of Lattice Polytopes
• Ehrhart polynomials of lattice polytopes with normalized volumes $5$
• A genetic algorithm to search the space of Ehrhart $h^{\ast }$-vectors
• Triangle Percolation on the Grid