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

## Synopsis

• Usage:
kreuzerSkarkeDim3()
• Outputs:
• a list, A list of KSEntry's, each one representing one of the 4319 polytopes (including description header information). Use matrix(KSEntry) to make any element in this list into something usable from Macaulay2.

## 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