April 30, 2002
We describe our package PALP of C programs for calculations with lattice polytopes and applications to toric geometry, which is freely available on the internet. It contains routines for vertex and facet enumeration, computation of incidences and symmetries, as well as completion of the set of lattice points in the convex hull of a given set of points. In addition, there are procedures specialised to reflexive polytopes such as the enumeration of reflexive subpolytopes, and applications to toric geometry and string theory, like the computation of Hodge data and fibration structures for toric Calabi-Yau varieties. The package is well tested and optimised in speed as it was used for time consuming tasks such as the classification of reflexive polyhedra in 4 dimensions and the creation and manipulation of very large lists of 5-dimensional polyhedra. While originally intended for low-dimensional applications, the algorithms work in any dimension and our key routine for vertex and facet enumeration compares well with existing packages.
Similar papers 1
June 22, 2011
We describe the C program mori.x. It is part of PALP, a package for analyzing lattice polytopes. Its main purpose is the construction and analysis of three--dimensional smooth Calabi--Yau hypersurfaces in toric varieties. The ambient toric varieties are given in terms of fans over the facets of reflexive lattice polytopes. The program performs crepant star triangulations of reflexive polytopes and determines the Mori cones of the resulting toric varieties. Furthermore, it com...
May 18, 2012
This article provides a complete user's guide to version 2.1 of the toric geometry package PALP by Maximilian Kreuzer and others. In particular, previously undocumented applications such as the program nef.x are discussed in detail. New features of PALP 2.1 include an extension of the program mori.x which can now compute Mori cones and intersection rings of arbitrary dimension and can also take specific triangulations of reflexive polytopes as input. Furthermore, the program ...
December 27, 1995
Reflexive polyhedra encode the combinatorial data for mirror pairs of Calabi-Yau hypersurfaces in toric varieties. We investigate the geometrical structures of circumscribed polytopes with a minimal number of facets and of inscribed polytopes with a minimal number of vertices. These objects, which constrain reflexive pairs of polyhedra from the interior and the exterior, can be described in terms of certain non-negative integral matrices. A major tool in the classification of...
February 28, 2000
Four dimensional reflexive polyhedra encode the data for smooth Calabi-Yau threefolds that are hypersurfaces in toric varieties, and have important applications both in perturbative and in non-perturbative string theory. We describe how we obtained all 473,800,776 reflexive polyhedra that exist in four dimensions and the 30,108 distinct pairs of Hodge numbers of the resulting Calabi-Yau manifolds. As a by-product we show that all these spaces (and hence the corresponding stri...
February 5, 2011
This is the first chapter in our "Toric Topology" book project. Further chapters are coming. Comments and suggestions are very welcome.
January 19, 2000
During the last years we have generated a large number of data related to Calabi-Yau hypersurfaces in toric varieties which can be described by reflexive polyhedra. We classified all reflexive polyhedra in three dimensions leading to K3 hypersurfaces and have nearly completed the four dimensional case relevant to Calabi-Yau threefolds. In addition, we have analysed for many of the resulting spaces whether they allow fibration structures of the types that are relevant in the c...
June 21, 2012
We provide a complete classification up to isomorphism of all smooth convex lattice 3-polytopes with at most 16 lattice points. There exist in total 103 different polytopes meeting these criteria. Of these, 99 are strict Cayley polytopes and the remaining 4 are obtained as inverse stellar subdivisions of such polytopes. We derive a classification, up to isomorphism, of all smooth embeddings of toric threefolds in $\mathbb{P}^N$ where $N\le 15$. Again we have in total 103 such...
October 31, 2014
In this paper we discuss combinatorial questions about lattice polytopes motivated by recent results on minimum distance estimation for toric codes. We also prove a new inductive bound for the minimum distance of generalized toric codes. As an application, we give new formulas for the minimum distance of generalized toric codes for special lattice point configurations.
June 11, 2012
The problem of counting points is revisited from the perspective of reflexive 4-dimensional polytopes. As an application, the Hilbert series of the 473,800,776 reflexive polytopes (equivalently, their Calabi-Yau hypersurfaces) are computed.
February 6, 2013
Toric geometry provides a bridge between the theory of polytopes and algebraic geometry: one can associate to each lattice polytope a polarized toric variety. In this paper we explore this correspondence to classify smooth lattice polytopes having small degree, extending a classification provided by Dickenstein, Di Rocco and Piene. Our approach consists in interpreting the degree of a polytope as a geometric invariant of the corresponding polarized variety, and then applying ...