Enumeration of lattice 3-polytopes by their number of lattice points

Abstract

We develop a procedure for the complete computational enumeration of lattice $3$-polytopes of width larger than one, up to any given number of lattice points. We also implement an algorithm for doing this and enumerate those with at most eleven lattice points (there are 216,453 of them). In order to achieve this we prove that if $P$ is a lattice 3-polytope of width larger than one and with at least seven lattice points then it fits in one of three categories that we call boxed, spiked and merged. Boxed polytopes have at most 11 lattice points; in particular they are finitely many, and we enumerate them completely with computer help. Spiked polytopes are infinitely many but admit a quite precise description (and enumeration). Merged polytopes are computed as a union (merging) of two polytopes of width larger than one and strictly smaller number of lattice points.

43 pages, 15 figures; Changes from v2: several editions, including title, many suggested by anonymous referees; added 4 figures; source and output files for the implemented algorithm can be found at http://personales.unican.es/santosf/3polytopes/

References in corpus (3)

• Lattice 3-polytopes with six lattice points
• Three-dimensional lattice polytopes with two interior lattice points
• Minimal volume $k$-point lattice $d$-simplices

Cited by in corpus (7)

• Lattice 3-polytopes with six lattice points
• Classification of triples of lattice polytopes with a given mixed volume
• Three-dimensional lattice polytopes with two interior lattice points
• Elementary moves on lattice polytopes
• Non-spanning lattice 3-polytopes
• The Finiteness Threshold Width of Lattice Polytopes
• The pyramidal growth