[Paper Review] On the enumeration of lattice $3$-polytopes

Mónica Blanco, Francisco Santos|arXiv (Cornell University)|Jan 11, 2016

Advanced Combinatorial Mathematics3 citations

TL;DR

This paper presents a complete classification of lattice 3-polytopes of width greater than one up to size 11, using three structural categories: spiked, boxed, and gluing-type polytopes. It proves that all such polytopes fall into one of these types, enabling full computational enumeration and completing the classification of distinct-pair-sum 3-polytopes up to size 8.

ABSTRACT

A lattice $3$-polytope is a polytope $P$ with integer vertices. We call size of $P$ the number of lattice points it contains, and width of $P$ the minimum, over all integer linear functionals $f$, of the length of the interval $f(P)$. In a previous paper we have shown that all but finitely many lattice $3$-polytopes of a given size have width one, which opens the possibility of enumerating those of width larger than one. There is none of size 4 (White, 1964) and those of sizes five and six were classified in our previous papers. In this paper we prove that every lattice $3$-polytope $P$ of width larger than one and size at least seven falls into one of the following three categories: - It projects in a very specific manner to one of a list of seven particular $2$-polytopes. We call $3$-polytopes of this type \emph{spiked}, and we describe them explicitly. - All except three of the lattice points in $P$ are contained in a rational parallelepiped of width one with respect to every facet. We call these polytopes \emph{boxed}. They have size at most 11 and we have completely enumerated them. - $P$ has (at least) two vertices $u$ and $v$ such that, when removing each of them, the width is still larger than one. Polytopes of this type can be all obtained gluing smaller polytopes of width larger than one. This allows for a computational enumeration of all lattice $3$-polytopes of width larger than one up to any given size. We have completely enumerated $3$-polytopes of width larger than one and size up to eleven. In particular, this implies the complete classification of distinct-pair-sum (or dps, for short) $3$-polytopes, which have size at most eight, of width larger than one.

Motivation & Objective

• To classify all lattice 3-polytopes of width greater than one up to a given size.
• To extend the classification of lattice 3-polytopes beyond size 6, building on prior work for sizes 4–6.
• To resolve the complete enumeration of distinct-pair-sum (dps) 3-polytopes, which are known to have size at most 8.
• To provide a structural decomposition of high-width lattice 3-polytopes into three canonical types: spiked, boxed, and gluing-based.
• To enable computational enumeration of all such polytopes up to size 11 by characterizing their geometric and combinatorial properties.

Proposed method

• Categorize lattice 3-polytopes of width >1 into three structural types: spiked, boxed, and gluing-based, based on their geometric and projection properties.
• Define 'spiked' polytopes as those projecting in a specific way to one of seven 2D lattice polytopes, allowing explicit description.
• Characterize 'boxed' polytopes as those where all but three lattice points lie within a rational parallelepiped of width one relative to each facet, with size at most 11.
• Use a gluing construction for polytopes with at least two vertices whose removal preserves width >1, enabling recursive enumeration.
• Implement computational enumeration by leveraging the structural decomposition, ensuring completeness up to size 11.
• Apply the classification to fully enumerate distinct-pair-sum (dps) 3-polytopes, which are known to be limited to size ≤8.

Experimental results

Research questions

• RQ1 What structural types characterize all lattice 3-polytopes of width greater than one and size at least seven?
• RQ2 How can lattice 3-polytopes of width >1 be systematically decomposed and enumerated using geometric and combinatorial criteria?
• RQ3 What is the maximum size of 'boxed' lattice 3-polytopes, and can they be completely enumerated?
• RQ4 Can all lattice 3-polytopes of width >1 be generated via gluing operations on smaller such polytopes?
• RQ5 What is the complete classification of distinct-pair-sum (dps) 3-polytopes, and up to what size is it achievable?

Key findings

• All lattice 3-polytopes of width >1 and size ≥7 fall into one of three structural categories: spiked, boxed, or gluing-based.
• The class of 'boxed' lattice 3-polytopes has size at most 11, and they have been completely enumerated in this work.
• The complete enumeration of lattice 3-polytopes of width >1 is achieved up to size 11, providing a full classification in this range.
• The classification of distinct-pair-sum (dps) 3-polytopes is fully resolved, as they are known to have size at most 8 and are included in the enumeration.
• The structural decomposition into spiked, boxed, and gluing types enables a complete computational enumeration of all such polytopes up to size 11.
• The paper establishes that no lattice 3-polytope of size 4 has width >1, confirming prior results and anchoring the classification framework.