Classification of empty lattice $4$-simplices of width larger than two

Abstract:

A lattice $d$-simplex is the convex hull of $d+1$ affinely independent integer points in ${\mathbb R}^d$. It is called empty if it contains no lattice point apart from its $d+1$ vertices. The classification of empty $3$-simplices has been known since 1964 (White), based on the fact that they all have width one. But for dimension $4$ no complete classification is known.

Haase and Ziegler (2000) enumerated all empty $4$-simplices up to determinant 1000 and based on their results conjectured that after determinant $179$ all empty $4$-simplices have width one or two. We prove this conjecture as follows:

- We show that no empty $4$-simplex of width three or more can have a determinant greater than 5058, by combining the recent classification of hollow 3-polytopes (Averkov, Krümpelmann and Weltge, 2017) with general methods from the geometry of numbers.

- We continue the computations of Haase and Ziegler up to determinant 7600, and find that no new $4$-simplices of width larger than two arise.

In particular, we give the whole list of empty $4$-simplices of width larger than two, which is as computed by Haase and Ziegler: There is a single empty $4$-simplex of width four (of determinant 101), and 178 empty $4$-simplices of width three, with determinants ranging from 41 to 179.