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

TL;DR

This paper presents a comprehensive computational method for enumerating lattice 3-polytopes with more than one width and up to eleven lattice points, classifying them into three categories and providing complete enumeration for the finite class.

Contribution

It introduces a novel classification of lattice 3-polytopes into boxed, spiked, and merged types, and develops algorithms for their enumeration and characterization.

Findings

Enumerated all lattice 3-polytopes with up to 11 points (216,453 total).

Proved that larger polytopes fit into three categories, enabling systematic enumeration.

Provided a complete computational framework for classifying and enumerating these polytopes.

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.