This method returns the cellular chain complex of $\Delta$ relative to its boundary. If $\Delta$ is homeomorphic to a disk, the homologies will vanish except for the top dimension.
i1 : V = {{0,0},{1,0},{0,1},{-1,-1}};
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i2 : F = {{0,1,2},{0,2,3},{0,1,3}};
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i3 : C = cellularComplex(V,F)
1 3 3
o3 = (QQ[t ..t ]) <-- (QQ[t ..t ]) <-- (QQ[t ..t ])
0 2 0 2 0 2
0 1 2
o3 : ChainComplex
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i4 : prune HH C
o4 = 0 : 0
1 : 0
1
2 : (QQ[t ..t ])
0 2
o4 : GradedModule
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If the complex is simplicial, there is no need for a vertex list.
i5 : F = {{0,1,2},{0,1,3},{0,2,3}};
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i6 : C = cellularComplex(F,InputType=>"Simplicial")
1 3 3
o6 = (QQ[t ..t ]) <-- (QQ[t ..t ]) <-- (QQ[t ..t ])
0 2 0 2 0 2
0 1 2
o6 : ChainComplex
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Arbitrary dimensions and polyhedral input are allowed.
i7 : V = {{1, 0, 0}, {-1, 0, 0}, {0, 1, 0}, {0, -1, 0}, {0, 0, 1}, {0, 0, -1}, {-2, -2, -2}, {-2, 2, -2}, {2, 2, -2}, {2, -2, -2}, {-2, -2, 2}, {-2, 2, 2}, {2, 2, 2}, {2, -2, 2}};
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i8 : F = {{0, 1, 2, 3, 4, 5}, {0, 8, 9, 12, 13}, {1, 6, 7, 10, 11}, {2, 7, 8, 11, 12}, {3, 6, 9, 10, 13}, {4, 10, 11, 12, 13}, {5, 6, 7, 8, 9}, {0, 2, 8, 12}, {0, 3, 9, 13}, {0, 4, 12, 13}, {0, 5, 8, 9}, {1, 2, 7, 11}, {1, 3, 6, 10}, {1, 4, 10, 11}, {1, 5, 6, 7}, {2, 4, 11, 12}, {3, 4, 10, 13}, {3, 5, 6, 9}, {2, 5, 7, 8}, {0, 2, 4, 12}, {0, 2, 5, 8}, {0, 3, 4, 13}, {0, 3, 5, 9}, {1, 2, 4, 11}, {1, 2, 5, 7}, {1, 3, 4, 10}, {1, 3, 5, 6}};
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i9 : C = cellularComplex(V,F); |
i10 : prune HH C
o10 = 0 : 0
1 : 0
2 : 0
1
3 : (QQ[t ..t ])
0 3
o10 : GradedModule
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The object cellularComplex is a method function with options.