981 research outputs found
Classification of empty lattice -simplices of width larger than two
A lattice -simplex is the convex hull of affinely independent
integer points in . It is called empty if it contains no lattice
point apart of its vertices. The classification of empty -simplices is
known since 1964 (White), based on the fact that they all have width one. But
for dimension no complete classification is known.
Haase and Ziegler (2000) enumerated all empty -simplices up to determinant
1000 and based on their results conjectured that after determinant all
empty -simplices have width one or two. We prove this conjecture as follows:
- We show that no empty -simplex of width three or more can have
determinant greater than 5058, by combining the recent classification of hollow
3-polytopes (Averkov, Kr\"umpelmann 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 -simplices of width larger than two arise.
In particular, we give the whole list of empty -simplices of width larger
than two, which is as computed by Haase and Ziegler: There is a single empty
-simplex of width four (of determinant 101), and 178 empty -simplices of
width three, with determinants ranging from 41 to 179.Comment: 21 pages, 5 figures; The appendix from v3 has been incorporated into
the main text. This version has been accepted for publication in Trans. Ame.
Math. So
Ehrhart polynomials of lattice polytopes with normalized volumes
A complete classification of the -vectors of lattice polytopes whose
normalized volumes are at most is known. In the present paper, we will
classify all the -vectors of lattice polytopes with normalized volumes
.Comment: 6 pages, to appear in Journal of Combinatoric
Gorenstein simplices with a given -polynomial
To classify the lattice polytopes with a given -polynomial is an
important open problem in Ehrhart theory. A complete classification of the
Gorenstein simplices whose normalized volumes are prime integers is known. In
particular, their -polynomials are of the form
, where and are positive integers. In the
present paper, a complete classification of the Gorenstein simplices with the
above -polynomials will be performed, when is either or ,
where and are prime integers with . Moreover, we consider the
number of Gorenstein simplices, up to unimodular equivalence, with the expected
-polynomial.Comment: 14 pages, to appear in Discrete Mathematic
Classification of lattice polytopes with small volumes
In the frame of a classification of general square systems of polynomial
equations solvable by radicals, Esterov and Gusev succeeded in classifying all
spanning lattice polytopes whose normalized volumes are at most . In the
present paper, we complete to classify all lattice polytopes whose normalized
volumes are at most based on the known classification of their
-polynomials.Comment: 12 pages, to appear in Journal of Combinatoric
A generalization of a theorem of G. K. White
An n-dimensional simplex \Delta in \R^n is called empty lattice simplex if
\Delta \cap\Z^n is exactly the set of vertices of \Delta . A theorem of G. K.
White shows that if n=3 then any empty lattice simplex \Delta \subset\R^3 is
isomorphic up to an unimodular affine linear transformation to a lattice
tetrahedron whose all vertices have third coordinate 0 or 1. In this paper we
prove a generalization of this theorem for an arbitrary odd dimension n=2d-1
which in some form was conjectured by Seb\H{o} and Borisov. This result implies
a classification of all 2d-dimensional isolated Gorenstein cyclic quotient
singularities with minimal log-discrepancy at least d.Comment: 12 pages, 2 figure
Gorenstein polytopes with trinomial -polynomials
The characterization of lattice polytopes based upon information about their
Ehrhart -polynomials is a difficult open problem. In this paper, we finish
the classification of lattice polytopes whose -polynomials satisfy two
properties: they are palindromic (so the polytope is Gorenstein) and they
consist of precisely three terms. This extends the classification of Gorenstein
polytopes of degree two due to Batyrev and Juny. The proof relies on the recent
characterization of Batyrev and Hofscheier of empty lattice simplices whose
-polynomials have precisely two terms. Putting our theorem in perspective,
we give a summary of these and other existing results in this area.Comment: 16 page
Non-spanning lattice 3-polytopes
We completely classify non-spanning -polytopes, by which we mean lattice
-polytopes whose lattice points do not affinely span the lattice. We show
that, except for six small polytopes (all having between five and eight lattice
points), every non-spanning -polytope has the following simple
description: consists of either (1) two lattice segments
lying in parallel and consecutive lattice planes or (2) a lattice segment
together with three or four extra lattice points placed in a very specific
manner. From this description we conclude that all the empty tetrahedra in a
non-spanning -polytope have the same volume and they form a
triangulation of , and we compute the -vectors of all non-spanning
-polytopes. We also show that all spanning -polytopes contain a
unimodular tetrahedron, except for two particular -polytopes with five
lattice points.Comment: 20 pages. Changes from v2: small changes requested by journal
referee; corrected typos in Thm 1.3; updated reference
Existence of unimodular triangulations - positive results
Unimodular triangulations of lattice polytopes arise in algebraic geometry,
commutative algebra, integer programming and, of course, combinatorics.
In this article, we review several classes of polytopes that do have
unimodular triangulations and constructions that preserve their existence.
We include, in particular, the first effective proof of the classical result
by Knudsen-Mumford-Waterman stating that every lattice polytope has a dilation
that admits a unimodular triangulation. Our proof yields an explicit (although
doubly exponential) bound for the dilation factor.Comment: 89 pages; changes from v2 and v1: the survey part has been expanded,
in particular the section on open question
Towards a mathematical formalism for classifying phases of matter
We propose a unified mathematical framework for classifying phases of matter.
The framework is based on different types of combinatorial structures with a
notion of locality called lattices. A tensor lattice is a local prescription
that associates tensor networks to those lattices. Different lattices are
related by local operations called moves. Those local operations define
consistency conditions for the tensors of the tensor network, the solutions to
which yield exactly solvable models for all kinds of phases. We implement the
framework to obtain models for symmetry-breaking and topological phases in up
to three space-time dimensions, their boundaries, defects, domain walls and
symmetries, as well as their anyons for 2+1-dimensional systems. We also
deliver ideas of how other kinds of phases, like SPT/SET, fermionic,
free-fermionic, chiral, and critical phases, can be described within our
framework. We also define another structure called contracted tensor lattices
which generalize tensor lattices: The former associate tensors instead of
tensor networks to lattices, and the consistency conditions for those tensors
are defined by another kind of local operation called gluings. Using this
generalization, our framework also covers mathematical structures like
axiomatic (non-fully extended or defective) TQFTs, that do not directly
describe phases on a microscopic physical level, but formalize certain aspects
of potential phases, like the anyon statistics of 2+1-dimensional phases. We
also introduce the very powerful concept of (contracted) tensor lattice
mapping, unifying a lots of different operations, such as stacking, anyon
fusion, anyon condensation, equivalence of different fixed point models, taking
the Drinfel'd centre, trivial defects or interpreting a bosonic model as a
fermionic model.Comment: First version of many mor
How to Classify Reflexive Gorenstein Cones
Two of my collaborations with Max Kreuzer involved classification problems
related to string vacua. In 1992 we found all 10,839 classes of polynomials
that lead to Landau-Ginzburg models with c=9 (Klemm and Schimmrigk also did
this); 7,555 of them are related to Calabi-Yau hypersurfaces. Later we found
all 473,800,776 reflexive polytopes in four dimensions; these give rise to
Calabi-Yau hypersurfaces in toric varieties. The missing piece - toric
constructions that need not be hypersurfaces - are the reflexive Gorenstein
cones introduced by Batyrev and Borisov. I explain what they are, how they
define the data for Witten's gauged linear sigma model, and how one can modify
our classification ideas to apply to them. I also present results on the first
and possibly most interesting step, the classification of certain basic weights
systems, and discuss limitations to a complete classification.Comment: 16 pages; contribution to the memorial volume `Strings, Gauge Fields,
and the Geometry Behind - The Legacy of Maximilian Kreuzer
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