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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.

Source code

The source code from which this documentation is derived is in the file ReflexivePolytopesDB.m2. The auxiliary files accompanying it are in the directory ReflexivePolytopesDB/.

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.