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

Abstract

[Abstract]: Rd. It is called empty if it contains no lattice point apart of its d + 1 vertices. The classification of empty 3-simplices is 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 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.