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ReflexivePolytopesDB -- simple access to Kreuzer-Skarke database of reflexive polytopes of dimensions 3 and 4

Description

In each given dimension $d$, it is known that the number of distinct (up to invertible integral change of basis) reflexive polytopes of dimension $d$ is finite in number. For example, in dimension 1 there is 1, in dimension 2, there are 16, in dimension 3, there are 4319 distinct reflexive polytopes.

In a major work, Max Kreuzer and Harold Skarke found algorithms for computing the set of such polytopes. They used these algorithms to show that there are 473,800,776 distinct 4-dimensional reflexive polytopes. The number is sufficiently large that they created a website http://hep.itp.tuwien.ac.at/~kreuzer/CY/ and an interface to access these examples. See their website for references to the algorithms used.

This package, ReflexivePolytopesDB, provides access to this database of reflexive polytopes of dimension 3 and dimension 4.

This package also contains a small part of this database for offline use, in case one cannot access the database.

Here we describe a simple use of the package. The actual investigation of the corresponding polytope or toric variety, or Calabi-Yau hypersurface, is done in Macaulay2 with the aid of other packages, such as Polyhedra and NormalToricVarieties.

Let's take one example polytope from the database, one whose corresponding Calabi-Yau 3-fold has Hodge numbers $h^{1,1}(X) = 9$ and $h^{1,2}(X) = 21$.

i1 : topes = kreuzerSkarke(9, 21);
using offline data file: ks9+21-n10.txt

This returns a list of single entries from the Kreuzer-Skarke database. Each one is essentially a string, containing a description line, together with the vertices of the corresponding polytope.

In Macaulay2, each entry is an object of class KSEntry (meaning: Kreuzer-Skarke database entry). Use matrix(KSEntry) to create the matrix whose columns are the vertices of the reflexive polytope. Use description(KSEntry) to see the associated description from the database (see Kreuzer-Skarke description headers for the description of the format of this description).

i2 : topes_1

o2 = 4 7  M:26 7 N:12 6 H:9,21 [-24] id:1
        1   1   0   0  -4   0   2
        0   3   0   0  -6   3   6
        0   0   1   1  -1  -2  -1
        0   0   0   2  -2  -1   1

o2 : KSEntry
 
i3 : A = matrix topes_1

o3 = | 1 1 0 0 -4 0  2  |
     | 0 3 0 0 -6 3  6  |
     | 0 0 1 1 -1 -2 -1 |
     | 0 0 0 2 -2 -1 1  |

              4       7
o3 : Matrix ZZ  <-- ZZ
 
i4 : description topes_1

o4 = 4 7  M:26 7 N:12 6 H:9,21 [-24] id:1

The corresponding reflexive polytope has 7 vertices, the columns of this matrix.

i5 : needsPackage "Polyhedra"

o5 = Polyhedra

o5 : Package
 
i6 : P = convexHull A

o6 = P

o6 : Polyhedron
 
i7 : assert isReflexive P
 
i8 : P2 = polar P

o8 = P2

o8 : Polyhedron
 
i9 : (numColumns vertices P, numColumns vertices P2)

o9 = (7, 6)

o9 : Sequence
 
i10 : (# latticePoints P, # latticePoints P2)

o10 = (26, 12)

o10 : Sequence

We can process many examples at one time, using the list facilities in Macaulay2. For instance, use List / Function to apply matrix to each element of the list, returning a list of the resulting matrices:

i11 : L = topes/matrix;
 
i12 : netList L

      +-----------------------------+
o12 = || 1 0 0 2 -4 |               |
      || 0 1 1 5 -7 |               |
      || 0 0 3 0 -3 |               |
      || 0 0 0 6 -6 |               |
      +-----------------------------+
      || 1 1 0 0 -4 0  2  |         |
      || 0 3 0 0 -6 3  6  |         |
      || 0 0 1 1 -1 -2 -1 |         |
      || 0 0 0 2 -2 -1 1  |         |
      +-----------------------------+
      || 1 0 0 -1 1 -1 |            |
      || 0 1 1 2  0 -4 |            |
      || 0 0 3 3  0 -6 |            |
      || 0 0 0 0  3 -3 |            |
      +-----------------------------+
      || 1 0 0 0 -3 -3 3  |         |
      || 0 1 1 0 2  0  -4 |         |
      || 0 0 3 0 3  0  -6 |         |
      || 0 0 0 1 -1 -2 2  |         |
      +-----------------------------+
      || 1 0 0 0 -2 -4 4  -4 |      |
      || 0 1 0 0 -2 -2 2  -3 |      |
      || 0 0 1 1 1  3  -5 2  |      |
      || 0 0 0 2 2  4  -6 3  |      |
      +-----------------------------+
      || 1 0 0 1 -2 -1 1  -1 -1 3  ||
      || 0 1 0 0 0  2  -2 0  0  -3 ||
      || 0 0 1 0 0  -1 1  1  -1 1  ||
      || 0 0 0 2 -1 -1 -1 -1 1  0  ||
      +-----------------------------+
      || 1 0 0 0 2  0  0  -1 -3 |   |
      || 0 1 1 0 -1 -1 -1 -1 3  |   |
      || 0 0 2 0 -1 -1 1  -2 0  |   |
      || 0 0 0 1 -1 1  -1 1  1  |   |
      +-----------------------------+
      || 1 0 1 0 -2 -1 -1 1  -1 3  ||
      || 0 1 0 0 1  2  0  -2 0  -3 ||
      || 0 0 2 0 -2 -1 -1 -1 1  0  ||
      || 0 0 0 1 0  -1 1  1  -1 1  ||
      +-----------------------------+
      || 1 0 0 -1 0 0 1  -4 3  -6 | |
      || 0 1 0 1  0 1 1  0  -1 -2 | |
      || 0 0 1 1  0 1 -2 4  -2 4  | |
      || 0 0 0 0  1 0 -1 1  -1 1  | |
      +-----------------------------+
      || 1 0 0 1  1  0 -2 4  -2 0  ||
      || 0 1 0 0  -1 0 3  -4 1  -2 ||
      || 0 0 1 -1 0  0 1  -4 3  -2 ||
      || 0 0 0 0  0  1 -1 1  -1 1  ||
      +-----------------------------+

Author

Version

This documentation describes version 1.0 of ReflexivePolytopesDB.

Citation

If you have used this package in your research, please cite it as follows:

@misc{ReflexivePolytopesDBSource,
  title = {{ReflexivePolytopesDB: A \emph{Macaulay2} package. Version~1.0}},
  author = {Mike Stillman},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/packages}}
}

Exports

  • Types
    • KSEntry -- an entry from the Kreuzer-Skarke database of dimension 3 and 4 reflexive polytopes
  • Functions and commands
    • availableOffline -- which Kreuzer-Skarke items are available offline
    • description -- the description header
    • generateOffline -- generate tables of reflexive 4d polytopes from the Kreuzer-Skarke list
    • kreuzerSkarke -- access Kreuzer-Skarke dimension 4 reflexive polytopes database
    • kreuzerSkarkeDim3 -- the list of 4319 dimension 3 reflexive polytopes in the Kreuzer-Skarke database
    • matrixFromString -- convert a string to a matrix of integers
    • onlineTests -- run a few tests which test access to the Kreuzer-Skarke database
  • Methods
    • description(KSEntry) -- see description -- the description header
    • generateOffline(ZZ) -- see generateOffline -- generate tables of reflexive 4d polytopes from the Kreuzer-Skarke list
    • generateOffline(ZZ,ZZ) -- see generateOffline -- generate tables of reflexive 4d polytopes from the Kreuzer-Skarke list
    • kreuzerSkarke(String) -- see kreuzerSkarke -- access Kreuzer-Skarke dimension 4 reflexive polytopes database
    • kreuzerSkarke(ZZ) -- see kreuzerSkarke -- access Kreuzer-Skarke dimension 4 reflexive polytopes database
    • kreuzerSkarke(ZZ,ZZ) -- see kreuzerSkarke -- access Kreuzer-Skarke dimension 4 reflexive polytopes database
    • matrix(KSEntry) -- convert a Kreuzer-Skarke entry to a matrix of integers
    • matrixFromString(String) -- see matrixFromString -- convert a string to a matrix of integers
    • toExternalString(KSEntry) -- a string suitable for input to Macaulay2
    • toString(KSEntry) -- the underlying string of a KSEntry
  • Symbols
    • Expected -- see generateOffline -- generate tables of reflexive 4d polytopes from the Kreuzer-Skarke list
    • Access -- see kreuzerSkarke -- access Kreuzer-Skarke dimension 4 reflexive polytopes database
    • DualLatticePoints -- see kreuzerSkarke -- access Kreuzer-Skarke dimension 4 reflexive polytopes database
    • Facets -- see kreuzerSkarke -- access Kreuzer-Skarke dimension 4 reflexive polytopes database
    • H12 -- see kreuzerSkarke -- access Kreuzer-Skarke dimension 4 reflexive polytopes database
    • LatticePoints -- see kreuzerSkarke -- access Kreuzer-Skarke dimension 4 reflexive polytopes database
    • Vertices -- see kreuzerSkarke -- access Kreuzer-Skarke dimension 4 reflexive polytopes database

For the programmer

The object ReflexivePolytopesDB is a package, defined in ReflexivePolytopesDB.m2, with auxiliary files in ReflexivePolytopesDB/.

The source of this document is in ReflexivePolytopesDB.m2:439:0.