kreuzerSkarke -- access Kreuzer-Skarke dimension 4 reflexive polytopes database

Synopsis

Usage:

kreuzerSkarke(h11, Limit=>n)

kreuzerSkarke(h11, h12)

kreuzerSkarke str
Inputs:
- h11, an integer, The desired Picard number of the Calabi-Yau hypersurface, a number between 1 and 491 inclusive.
- h12, an integer, (optional) the desired $h^{1,2}$ of the Calabi-Yau hypersurface
- str, a string, parses the given string into a list of KSEntry's
Optional inputs:
- Limit (missing documentation) => an integer, default value 1000, The maximum number of examples to retrieve
- Vertices (missing documentation) => an integer, default value null, Restrict to those examples with this number of vertices
- Facets (missing documentation) => an integer, default value null, Restrict to those examples with this number of facets
- LatticePoints (missing documentation) => an integer, default value null, Restrict to those examples with this number of lattice points
- DualLatticePoints (missing documentation) => an integer, default value null, Restrict to those examples whose polar dual has this number of lattice points
- H12 (missing documentation) => an integer, default value null, Restrict to those examples whose associated Calabi-Yau 3-fold has the given $h^{1,2}(X)$
- Access (missing documentation) => a string, default value null, Either null (use Macaulay2's getWWW), "wget" or "curl". Mainly an internal optional argument for developers
Outputs:
- a list, of KSEntry's, each of which is one entry (one polytope) from the database. Use matrix(KSEntry) and description(KSEntry) to extract information from an entry.

Description

As a an example, let's take the 4th example with $h^{11}=5$, $h^{21}=53$.

i1 : topes = kreuzerSkarke(5, 53, Limit=>4)
using offline data file: ks5+53-n204.txt

o1 = {4 5  M:51 5 N:11 5 H:5,53 [-96] id:0, 4 5  M:61 5 N:9 5 H:5,53 [-96]
         1   0   1   1  -8                     1   0   0   4  -8          
         0   1   0   0  -3                     0   1   1   9 -15          
         0   0   3   3 -12                     0   0   2   0  -6          
         0   0   0   6 -12                     0   0   0  12 -12          
     ------------------------------------------------------------------------
     id:1, 4 9  M:60 9 N:10 6 H:5,53 [-96] id:2, 4 9  M:59 9 N:10 6 H:5,53
              1   1   1   1   0  -1  -1  -2  -4     1   1   1   0   0   0 
              0   3   0   1   0  -1  -4   0  -4     0   2   0   0   4   0 
              0   0   2   2   0   2   2  -6  -4     0   0   2   0   0   4 
              0   0   0   0   1  -2  -2   3   1     0   0   0   1   1   1 
     ------------------------------------------------------------------------
     [-96] id:3}
     -2  -2  -2
     -4  -6  -6
     -6  -6  -4
     -3  -3  -3

o1 : List

i2 : assert(#topes == 4)

i3 : tope = topes_3

o3 = 4 9  M:59 9 N:10 6 H:5,53 [-96] id:3
        1   1   1   0   0   0  -2  -2  -2
        0   2   0   0   4   0  -4  -6  -6
        0   0   2   0   0   4  -6  -6  -4
        0   0   0   1   1   1  -3  -3  -3

o3 : KSEntry

i4 : header = description tope

o4 = 4 9  M:59 9 N:10 6 H:5,53 [-96] id:3

i5 : A = matrix tope

o5 = | 1 1 1 0 0 0 -2 -2 -2 |
     | 0 2 0 0 4 0 -4 -6 -6 |
     | 0 0 2 0 0 4 -6 -6 -4 |
     | 0 0 0 1 1 1 -3 -3 -3 |

              4       9
o5 : Matrix ZZ  <-- ZZ

The description line gives some information about the example, see Kreuzer-Skarke description headers for more details. The polytope is the convex hull of the columns of the matrix $A$.

Some of the Kreuzer-Skarke data is available from this package off-line. To see what is available when not connected to the net, or when the website is down, call availableOffline.

i6 : availableOffline()

o6 = HashTable{"kreuzerSkarke(1)" => /usr/local/share/Macaulay2/ReflexivePolytopesDB/ks1-n5.txt                  }
               "kreuzerSkarke(2)" => /usr/local/share/Macaulay2/ReflexivePolytopesDB/ks2-n36.txt
               "kreuzerSkarke(3)" => /usr/local/share/Macaulay2/ReflexivePolytopesDB/ks3-n244.txt
               "kreuzerSkarke(5, Limit => 50)" => /usr/local/share/Macaulay2/ReflexivePolytopesDB/ks5-n50.txt
               "kreuzerSkarke(5,53)" => /usr/local/share/Macaulay2/ReflexivePolytopesDB/ks5+53-n204.txt
               "kreuzerSkarke(9,21)" => /usr/local/share/Macaulay2/ReflexivePolytopesDB/ks9+21-n10.txt
               "kreuzerSkarke(11,24)" => /usr/local/share/Macaulay2/ReflexivePolytopesDB/ks11+24-n200.txt
               "kreuzerSkarke(21, Limit => 100)" => /usr/local/share/Macaulay2/ReflexivePolytopesDB/ks21-n100.txt
               "kreuzerSkarke(300)" => /usr/local/share/Macaulay2/ReflexivePolytopesDB/ks300-n20.txt
               "kreuzerSkarke(491)" => /usr/local/share/Macaulay2/ReflexivePolytopesDB/ks491-n1.txt

o6 : HashTable

If data can be obtained locally, it will, and a message indicating this is output.

If one has a string containing Kreuzer-Skarke entries, one can use this function to make a list of KSEntry's

i7 : str = "4 12  M:34 12 N:17 8 H:17,30 [-26] id:0
       1   1   1   1   0   0   0  -3  -3  -1  -1  -3
       0   3   0   1   0   1   0  -3  -5   0  -1  -3
       0   0   2   2   0   0   1  -2  -2  -1  -1  -2
       0   0   0   0   1   1   1  -1  -1   1   1   1"

o7 = 4 12  M:34 12 N:17 8 H:17,30 [-26] id:0
       1   1   1   1   0   0   0  -3  -3  -1  -1  -3
       0   3   0   1   0   1   0  -3  -5   0  -1  -3
       0   0   2   2   0   0   1  -2  -2  -1  -1  -2
       0   0   0   0   1   1   1  -1  -1   1   1   1

i8 : L = kreuzerSkarke str

o8 = {4 12  M:34 12 N:17 8 H:17,30 [-26] id:0 id:0   }
        1   1   1   1   0   0   0  -3  -3  -1  -1  -3
        0   3   0   1   0   1   0  -3  -5   0  -1  -3
        0   0   2   2   0   0   1  -2  -2  -1  -1  -2
        0   0   0   0   1   1   1  -1  -1   1   1   1

o8 : List

i9 : matrix first L

o9 = | 1 1 1 1 0 0 0 -3 -3 -1 -1 -3 |
     | 0 3 0 1 0 1 0 -3 -5 0  -1 -3 |
     | 0 0 2 2 0 0 1 -2 -2 -1 -1 -2 |
     | 0 0 0 0 1 1 1 -1 -1 1  1  1  |

              4       12
o9 : Matrix ZZ  <-- ZZ

Caveat

The database is organized by the h11 value. Placing filters can significantly slow down the database access. Using any of the optional arguments other than Limit also means that the locally stored datafiles will not be used.

Ways to use kreuzerSkarke :

kreuzerSkarke(String)
kreuzerSkarke(ZZ)
kreuzerSkarke(ZZ,ZZ)

For the programmer

The object kreuzerSkarke is a method function with options.