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ReflexivePolytopesDB :: kreuzerSkarkeDim3

kreuzerSkarkeDim3 -- the list of 4319 dimension 3 reflexive polytopes in the Kreuzer-Skarke database

Synopsis

Description

As a an example, let’s take the 101th example on this list.

i1 : topes = kreuzerSkarkeDim3();
i2 : #topes

o2 = 4319
i3 : tope = topes_100

o3 = 3 5  M:12 5 N:15 5 Pic:13 Cor:4 id:100
         1    0    0   -3    1
         0    1    0    0   -2
         0    0    1   -2    2

o3 : KSEntry
i4 : header = description tope

o4 = 3 5  M:12 5 N:15 5 Pic:13 Cor:4 id:100
i5 : A = matrix tope

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

              3        5
o5 : Matrix ZZ  <--- ZZ

The first 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.

One can use the packages Polyhedra and NormalToricVarieties to investigate these polyhedra, and the associated toric varieties.

i6 : needsPackage "Polyhedra"

o6 = Polyhedra

o6 : Package
i7 : P = convexHull A

o7 = P

o7 : Polyhedron
i8 : P2 = polar P

o8 = P2

o8 : Polyhedron
i9 : # latticePoints P

o9 = 12
i10 : # latticePoints P2

o10 = 15
i11 : # vertices P

o11 = 5
i12 : # vertices P2

o12 = 5
i13 : isReflexive P

o13 = true
i14 : needsPackage "NormalToricVarieties"

o14 = NormalToricVarieties

o14 : Package
i15 : V0 = normalToricVariety normalFan P

o15 = V0

o15 : NormalToricVariety
i16 : dim V0

o16 = 3
i17 : max V0

o17 = {{0, 1, 2}, {0, 1, 4}, {0, 2, 3, 4}, {1, 2, 3}, {1, 3, 4}}

o17 : List
i18 : rays V0

o18 = {{1, 0, -1}, {-1, -1, -1}, {1, -1, -1}, {-1, -1, 2}, {-1, 2, 2}}

o18 : List
i19 : V = makeSimplicial V0

o19 = V

o19 : NormalToricVariety
i20 : isSimplicial V

o20 = true
i21 : isProjective V

o21 = true
i22 : isSmooth V

o22 = false
i23 : dim V

o23 = 3

See also