Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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


Authors: Óscar Iglesias-Valiño and Francisco Santos
Journal: Trans. Amer. Math. Soc. 371 (2019), 6605-6625
MSC (2010): Primary 52B20; Secondary 11H06
DOI: https://doi.org/10.1090/tran/7531
Published electronically: January 24, 2019
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 52B20, 11H06

Retrieve articles in all journals with MSC (2010): 52B20, 11H06


Additional Information

Óscar Iglesias-Valiño
Affiliation: Department of Mathematics, Statistics and Computer Science. University of Cantabria, Cantabria, Spain
Email: oscar.iglesias@unican.es

Francisco Santos
Affiliation: Department of Mathematics, Statistics and Computer Science. University of Cantabria, Cantabria, Spain
Email: francisco.santos@unican.es

DOI: https://doi.org/10.1090/tran/7531
Keywords: Lattice polytopes, empty, classification, simplices, dimension 4, lattice width, maximum volume, enumeration.
Received by editor(s): May 5, 2017
Received by editor(s) in revised form: July 14, 2017, and February 2, 2018
Published electronically: January 24, 2019
Additional Notes: This work was supported by grants MTM2014-54207-P (both authors) and BES-2015-073128 (first author) of the Spanish Ministry of Economy and Competitiveness.
The second author was also supported by an Einstein Visiting Professorship of the Einstein Foundation Berlin
Article copyright: © Copyright 2019 American Mathematical Society