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### 16 A demo session with simpcomp

This chapter contains the transcript of a demo session with simpcomp that is intended to give an insight into what things can be done with this package.

Of course this only scratches the surface of the functions provided by simpcomp. See Chapters 4 through 14 for further functions provided by simpcomp.

#### 16.1 Creating a `SCSimplicialComplex` object

Simplicial complex objects can either be created from a facet list (complex `c1` below), orbit representatives together with a permutation group (complex `c2`) or difference cycles (complex `c3`, see Section 6.1), from a function generating triangulations of standard complexes (complex `c4`, see Section 6.2) or from a function constructing infinite series for combinatorial (pseudo)manifolds (complexes `c5`, `c6`, `c7`, see Section 6.3 and the function prefix `SCSeries...`). There are also functions creating new simplicial complexes from old, see Section 6.5, which will be described in the next sections.

```gap> #first run functionality test on simpcomp

gap> SCRunTest();

+ test simpcomp package, version 1.6.1
+ GAP4stones: 69988
true
gap> #all ok

gap> c1:=SCFromFacets([[1,2],[2,3],[3,1]]);

[SimplicialComplex

Properties known: Dim, Facets, Name, VertexLabels.

Name="unnamed complex 1"
Dim=1

/SimplicialComplex]
gap> G:=Group([(2,12,11,6,8,3)(4,7,10)(5,9),(1,11,6,4,5,3,10,8,9,7,2,12)]);

Group([ (2,12,11,6,8,3)(4,7,10)(5,9), (1,11,6,4,5,3,10,8,9,7,2,12) ])
gap> StructureDescription(G);

"S4 x S3"
gap> Size(G);

144
gap> c2:=SCFromGenerators(G,[[1,2,3]]);;

gap> c2.IsManifold;

true
gap> SCLibDetermineTopologicalType(c2);

[SimplicialComplex

Properties known: AutomorphismGroup, AutomorphismGroupSize,
AutomorphismGroupStructure, AutomorphismGroupTransitivity,\

Boundary, Dim, Faces, Facets, Generators, HasBoundary,
IsManifold, IsPM, Name, TopologicalType, VertexLabels,
Vertices.

Name="complex from generators under group S4 x S3"
Dim=2
AutomorphismGroupSize=144
AutomorphismGroupStructure="S4 x S3"
AutomorphismGroupTransitivity=1
HasBoundary=false
IsPM=true
TopologicalType="T^2"

/SimplicialComplex]
gap> c3:=SCFromDifferenceCycles([[1,1,6],[3,3,2]]);

[SimplicialComplex

Properties known: Dim, Facets, Name, VertexLabels.

Name="complex from diffcycles [ [ 1, 1, 6 ], [ 3, 3, 2 ] ]"
Dim=2

/SimplicialComplex]
gap> c4:=SCBdSimplex(2);

[SimplicialComplex

Properties known: AutomorphismGroup, AutomorphismGroupOrder,
AutomorphismGroupStructure, AutomorphismGroupTransitivity,
Chi, Dim, F, Facets, Generators, HasBounday, Homology,
IsConnected, IsStronglyConnected, Name, TopologicalType,
VertexLabels.

Name="S^1_3"
Dim=1
AutomorphismGroupStructure="S3"
AutomorphismGroupTransitivity=3
Chi=0
F=[ 3, 3 ]
Homology=[ [ 0, [ ] ], [ 1, [ ] ] ]
IsConnected=true
IsStronglyConnected=true
TopologicalType="S^1"

/SimplicialComplex]
gap> c5:=SCSeriesCSTSurface(2,16);;

gap> SCLibDetermineTopologicalType(c5);

[SimplicialComplex

Properties known: Boundary, Dim, Faces, Facets, HasBoundary, IsPM, Name,
TopologicalType, VertexLabels.

Name="cst surface S_{(2,16)} = { (2:2:12),(6:6:4) }"
Dim=2
HasBoundary=false
IsPM=true
TopologicalType="T^2 U T^2"

/SimplicialComplex]
gap> c6:=SCSeriesD2n(22);;

gap> c6.Homology;

[ [ 0, [  ] ], [ 1, [  ] ], [ 0, [ 2 ] ], [ 0, [  ] ] ]
gap> c6.F;

[ 44, 264, 440, 220 ]
gap> SCSeriesAGL(17);

[ AGL(1,17), [ [ 1, 2, 4, 8, 16 ] ] ]
gap> c7:=SCFromGenerators(last[1],last[2]);;

gap> c7.AutomorphismGroupTransitivity;

2
```

#### 16.2 Working with a `SCSimplicialComplex` object

As described in Section 3.1 there are two several ways of accessing an object of type `SCSimplicialComplex`. An example for the two equivalent ways is given below. The preference will be given to the object oriented notation in this demo session. The code listed below

```gap> c:=SCBdSimplex(3);; # create a simplicial complex object

gap> SCFVector(c);

[ 4, 6, 4 ]
gap> SCSkel(c,0);

[ [ 1 ], [ 2 ], [ 3 ], [ 4 ] ]
```

is equivalent to

```gap> c:=SCBdSimplex(3);; # create a simplicial complex object

gap> c.F;

[ 4, 6, 4 ]
gap> c.Skel(0);

[ [ 1 ], [ 2 ], [ 3 ], [ 4 ] ]
```

#### 16.3 Calculating properties of a `SCSimplicialComplex` object

simpcomp provides a variety of functions for calculating properties of simplicial complexes, see Section 6.8. All these properties are only calculated once and stored in the `SCSimplicialComplex` object.

```gap> c1.F;

[ 3, 3 ]
gap> c1.FaceLattice;

[ [ [ 1 ], [ 2 ], [ 3 ] ], [ [ 1, 2 ], [ 1, 3 ], [ 2, 3 ] ] ]
gap> c1.AutomorphismGroup;

S3
gap> c1.Generators;

[ [ [ 1, 2 ], 3 ] ]
gap> c3.Facets;

[ [ 1, 2, 3 ], [ 1, 2, 8 ], [ 1, 3, 6 ], [ 1, 4, 6 ], [ 1, 4, 7 ],
[ 1, 7, 8 ], [ 2, 3, 4 ], [ 2, 4, 7 ], [ 2, 5, 7 ], [ 2, 5, 8 ],
[ 3, 4, 5 ], [ 3, 5, 8 ], [ 3, 6, 8 ], [ 4, 5, 6 ], [ 5, 6, 7 ],
[ 6, 7, 8 ] ]
gap> c3.F;

[ 8, 24, 16 ]
gap> c3.G;

[ 4 ]
gap> c3.H;

[ 5, 11, -1 ]
gap> c3.ASDet;

186624
gap> c3.Chi;

0
gap> c3.Generators;

[ [ [ 1, 2, 3 ], 16 ] ]
gap> c3.HasBoundary;

false
gap> c3.IsConnected;

true
gap> c3.IsCentrallySymmetric;

true
gap> c3.Vertices;

[ 1, 2, 3, 4, 5, 6, 7, 8 ]
gap> c3.ConnectedComponents;

[ [SimplicialComplex

Properties known: Dim, Facets, Name, VertexLabels.

Name="Connected component #1 of complex from diffcycles [ [ 1, 1, 6 ], [ \
3, 3, 2 ] ]"
Dim=2

/SimplicialComplex] ]
gap> c3.UnknownProperty;

#I  SCPropertyObject: unhandled property 'UnknownProperty'. Handled properties\
are [ "Equivalent", "IsKStackedSphere", "IsManifold", "IsMovable", "Move",
"Moves", "RMoves", "ReduceAsSubcomplex", "Reduce", "ReduceEx", "Copy",
"Recalc", "ASDet", "AutomorphismGroup", "AutomorphismGroupInternal",
"Boundary", "ConnectedComponents", "Dim", "DualGraph", "Chi", "F",
"FaceLattice", "FaceLatticeEx", "Faces", "FacesEx", "Facets", "FacetsEx",
"FpBetti", "FundamentalGroup", "G", "Generators", "GeneratorsEx", "H",
"HasBoundary", "HasInterior", "Homology", "Incidences", "IncidencesEx",
"Interior", "IsCentrallySymmetric", "IsConnected", "IsEmpty",
"IsEulerianManifold", "IsHomologySphere", "IsInKd", "IsKNeighborly",
"IsOrientable", "IsPM", "IsPure", "IsShellable", "IsStronglyConnected",
"MinimalNonFaces", "MinimalNonFacesEx", "Name", "Neighborliness",
"Orientation", "Skel", "SkelEx", "SpanningTree",
"StronglyConnectedComponents", "Vertices", "VerticesEx",
"BoundaryOperatorMatrix", "HomologyBasis", "HomologyBasisAsSimplices",
"HomologyInternal", "CoboundaryOperatorMatrix", "Cohomology",
"CohomologyBasis", "CohomologyBasisAsSimplices", "CupProduct",
"IntersectionForm", "IntersectionFormParity",
"ExportLatexTable", "ExportJavaView", "LabelMax", "LabelMin", "Labels",
"Relabel", "RelabelStandard", "RelabelTransposition", "Rename",
"SortComplex", "UnlabelFace", "AlexanderDual", "CollapseGreedy", "Cone",
"IsIsomorphic", "IsSubcomplex", "Isomorphism", "IsomorphismEx", "Join",
"Shellings", "Span", "Star", "Stars", "Suspension", "Union",
"VertexIdentification", "Wedge", "DetermineTopologicalType", "Dim",
"Facets", "VertexLabels", "Name", "Vertices", "IsConnected",
"ConnectedComponents" ].

fail
```

#### 16.4 Creating new complexes from a `SCSimplicialComplex` object

As already mentioned, there is the possibility to generate new objects of type `SCSimplicialComplex` from existing ones using standard constructions. The functions used in this section are described in more detail in Section 6.5.

```gap> d:=c3+c3;

[SimplicialComplex

Properties known: Dim, Facets, Name, VertexLabels, Vertices.

Name="complex from diffcycles [ [ 1, 1, 6 ], [ 3, 3, 2 ] ]#+-complex from dif\
fcycles [ [ 1, 1, 6 ], [ 3, 3, 2 ] ]"
Dim=2

/SimplicialComplex]
gap> SCRename(d,"T^2#T^2");

true

[SimplicialComplex

Properties known: Dim, Facets, Name, VertexLabels.

Name="lk(1) in T^2#T^2"
Dim=1

/SimplicialComplex]
gap> SCStar(d,[1,2]);

[SimplicialComplex

Properties known: Dim, Facets, Name, VertexLabels.

Name="star([ 1, 2 ]) in T^2#T^2"
Dim=2

/SimplicialComplex]
gap> SCRename(c3,"T^2");

true
gap> SCConnectedProduct(c3,4);

[SimplicialComplex

Properties known: Dim, Facets, Name, VertexLabels, Vertices.

Name="T^2#+-T^2#+-T^2#+-T^2"
Dim=2

/SimplicialComplex]
gap> SCCartesianProduct(c4,c4);

[SimplicialComplex

Properties known: Dim, Facets, Name, TopologicalType, VertexLabels.

Name="S^1_3xS^1_3"
Dim=2
TopologicalType="S^1xS^1"

/SimplicialComplex]
gap> SCCartesianPower(c4,3);

[SimplicialComplex

Properties known: Dim, Facets, Name, TopologicalType, VertexLabels.

Name="(S^1_3)^3"
Dim=3
TopologicalType="(S^1)^3"

/SimplicialComplex]
```

#### 16.5 Homology related calculations

simpcomp relies on the GAP package homology [DHSW04] for its homology computations but provides further (co-)homology related functions, see Chapter 8.

```gap> s2s2:=SCCartesianProduct(SCBdSimplex(3),SCBdSimplex(3));

[SimplicialComplex

Properties known: Dim, Facets, Name, TopologicalType, VertexLabels.

Name="S^2_4xS^2_4"
Dim=4
TopologicalType="S^2xS^2"

/SimplicialComplex]
gap> SCHomology(s2s2);

[ [ 0, [  ] ], [ 0, [  ] ], [ 2, [  ] ], [ 0, [  ] ], [ 1, [  ] ] ]
gap> SCHomologyInternal(s2s2);

[ [ 0, [  ] ], [ 0, [  ] ], [ 2, [  ] ], [ 0, [  ] ], [ 1, [  ] ] ]
gap> SCHomologyBasis(s2s2,2);

[ [ 1, [ [ 1, 70 ], [ -1, 12 ], [ 1, 2 ], [ -1, 1 ] ] ],
[ 1, [ [ 1, 143 ], [ -1, 51 ], [ 1, 29 ], [ -1, 25 ] ] ] ]
gap> SCHomologyBasisAsSimplices(s2s2,2);

[ [ 1,
[ [ 1, [ 2, 3, 4 ] ], [ -1, [ 1, 3, 4 ] ], [ 1, [ 1, 2, 4 ] ], [ -1, [ 1
, 2, 3 ] ] ] ],
[ 1, [ [ 1, [ 5, 9, 13 ] ], [ -1, [ 1, 9, 13 ] ], [ 1, [ 1, 5, 13 ] ],
[ -1, [ 1, 5, 9 ] ] ] ] ]
gap> SCCohomologyBasis(s2s2,2);

[ [ 1,
[ [ 1, 122 ], [ 1, 115 ], [ 1, 112 ], [ 1, 111 ], [ 1, 93 ], [ 1, 90 ],
[ 1, 89 ], [ 1, 84 ], [ 1, 83 ], [ 1, 82 ], [ 1, 46 ], [ 1, 43 ],
[ 1, 42 ], [ 1, 37 ], [ 1, 36 ], [ 1, 35 ], [ 1, 28 ], [ 1, 27 ],
[ 1, 26 ], [ 1, 25 ] ] ],
[ 1, [ [ 1, 213 ], [ 1, 201 ], [ 1, 192 ], [ 1, 189 ], [ 1, 159 ],
[ 1, 150 ], [ 1, 147 ], [ 1, 131 ], [ 1, 128 ], [ 1, 125 ],
[ 1, 67 ], [ 1, 58 ], [ 1, 55 ], [ 1, 39 ], [ 1, 36 ], [ 1, 33 ],
[ 1, 10 ], [ 1, 7 ], [ 1, 4 ], [ 1, 1 ] ] ] ]
gap> SCCohomologyBasisAsSimplices(s2s2,2);

[ [ 1, [ [ 1, [ 4, 8, 12 ] ], [ 1, [ 3, 8, 12 ] ], [ 1, [ 3, 7, 12 ] ],
[ 1, [ 3, 7, 11 ] ], [ 1, [ 2, 8, 12 ] ], [ 1, [ 2, 7, 12 ] ],
[ 1, [ 2, 7, 11 ] ], [ 1, [ 2, 6, 12 ] ], [ 1, [ 2, 6, 11 ] ],
[ 1, [ 2, 6, 10 ] ], [ 1, [ 1, 8, 12 ] ], [ 1, [ 1, 7, 12 ] ],
[ 1, [ 1, 7, 11 ] ], [ 1, [ 1, 6, 12 ] ], [ 1, [ 1, 6, 11 ] ],
[ 1, [ 1, 6, 10 ] ], [ 1, [ 1, 5, 12 ] ], [ 1, [ 1, 5, 11 ] ],
[ 1, [ 1, 5, 10 ] ], [ 1, [ 1, 5, 9 ] ] ] ],
[ 1, [ [ 1, [ 13, 14, 15 ] ], [ 1, [ 9, 14, 15 ] ], [ 1, [ 9, 10, 15 ] ],
[ 1, [ 9, 10, 11 ] ], [ 1, [ 5, 14, 15 ] ], [ 1, [ 5, 10, 15 ] ],
[ 1, [ 5, 10, 11 ] ], [ 1, [ 5, 6, 15 ] ], [ 1, [ 5, 6, 11 ] ],
[ 1, [ 5, 6, 7 ] ], [ 1, [ 1, 14, 15 ] ], [ 1, [ 1, 10, 15 ] ],
[ 1, [ 1, 10, 11 ] ], [ 1, [ 1, 6, 15 ] ], [ 1, [ 1, 6, 11 ] ],
[ 1, [ 1, 6, 7 ] ], [ 1, [ 1, 2, 15 ] ], [ 1, [ 1, 2, 11 ] ],
[ 1, [ 1, 2, 7 ] ], [ 1, [ 1, 2, 3 ] ] ] ] ]
gap> PrintArray(SCIntersectionForm(s2s2));

[ [  0,  1 ],
[  1,  0 ] ]
gap> c:=s2s2+s2s2;

[SimplicialComplex

Properties known: Dim, Facets, Name, VertexLabels, Vertices.

Name="S^2_4xS^2_4#+-S^2_4xS^2_4"
Dim=4

/SimplicialComplex]
gap> PrintArray(SCIntersectionForm(c));

[ [   0,  -1,   0,   0 ],
[  -1,   0,   0,   0 ],
[   0,   0,   0,  -1 ],
[   0,   0,  -1,   0 ] ]
```

#### 16.6 Bistellar flips

For a more detailed description of functions related to bistellar flips as well as a very short introduction into the topic, see Chapter 9.

```gap> beta4:=SCBdCrossPolytope(4);;

gap> s3:=SCBdSimplex(4);;

gap> SCEquivalent(beta4,s3);

#I  round 0, move: [ [ 2, 6, 7 ], [ 3, 4 ] ]
[ 8, 25, 34, 17 ]
#I  round 1, move: [ [ 2, 7 ], [ 3, 4, 5 ] ]
[ 8, 24, 32, 16 ]
#I  round 2, move: [ [ 2, 5 ], [ 3, 4, 8 ] ]
[ 8, 23, 30, 15 ]
#I  round 3, move: [ [ 2 ], [ 3, 4, 6, 8 ] ]
[ 7, 19, 24, 12 ]
#I  round 4, move: [ [ 6, 8 ], [ 1, 3, 4 ] ]
[ 7, 18, 22, 11 ]
#I  round 5, move: [ [ 8 ], [ 1, 3, 4, 5 ] ]
[ 6, 14, 16, 8 ]
#I  round 6, move: [ [ 5 ], [ 1, 3, 4, 7 ] ]
[ 5, 10, 10, 5 ]
#I  SCReduceComplexEx: complexes are bistellarly equivalent.
true
gap> SCBistellarOptions.WriteLevel;

0
gap> SCBistellarOptions.WriteLevel:=1;

1
gap> SCEquivalent(beta4,s3);

#I  SCLibInit: made directory "~/PATH" for user library.
#I  SCLibUpdate: rebuilding index for ~/PATH.
#I  SCLibUpdate: rebuilding index done.

#I  round 0, move: [ [ 2, 4, 6 ], [ 7, 8 ] ]
[ 8, 25, 34, 17 ]
#I  round 1, move: [ [ 2, 4 ], [ 5, 7, 8 ] ]
[ 8, 24, 32, 16 ]
#I  round 2, move: [ [ 4, 5 ], [ 1, 7, 8 ] ]
[ 8, 23, 30, 15 ]
#I  round 3, move: [ [ 4 ], [ 1, 6, 7, 8 ] ]
[ 7, 19, 24, 12 ]
#I  SCLibAdd: saving complex to file "complex_ReducedComplex_7_vertices_3_2009\
-10-27_11-40-00.sc".
#I  round 4, move: [ [ 2, 6 ], [ 3, 7, 8 ] ]
[ 7, 18, 22, 11 ]
#I  round 5, move: [ [ 2 ], [ 3, 5, 7, 8 ] ]
[ 6, 14, 16, 8 ]
#I  SCLibAdd: saving complex to file "complex_ReducedComplex_6_vertices_5_2009\
-10-27_11-40-00.sc".
#I  round 6, move: [ [ 5 ], [ 1, 3, 7, 8 ] ]
[ 5, 10, 10, 5 ]
#I  SCLibAdd: saving complex to file "complex_ReducedComplex_5_vertices_6_2009\
-10-27_11-40-00.sc".
#I  SCLibAdd: saving complex to file "complex_ReducedComplex_5_vertices_7_2009\
-10-27_11-40-00.sc".
#I  SCReduceComplexEx: complexes are bistellarly equivalent.
true
gap> myLib:=SCLibInit("~/PATH"); # copy path from above

[Simplicial complex library. Properties:
CalculateIndexAttributes=true
Number of complexes in library=4
IndexAttributes=[ "Name", "Date", "Dim", "F", "G", "H", "Chi", "Homology" ]
Path="/home/spreerjn/reducedComplexes/2009-10-27_11-40-00/"
]

[SimplicialComplex

Properties known: Chi, Date, Dim, F, Faces, Facets, G, H, Homology,
IsConnected, Name, VertexLabels.

Name="ReducedComplex_5_vertices_6"
Dim=3
Chi=0
F=[ 5, 10, 10, 5 ]
G=[ 0, 0 ]
H=[ 1, 1, 1, 1 ]
Homology=[ [ 0, [ ] ], [ 0, [ ] ], [ 0, [ ] ], [ 1, [ ] ] ]
IsConnected=true

/SimplicialComplex]

[SimplicialComplex

Properties known: Chi, Date, Dim, F, Faces, Facets, G, H, Homology,
IsConnected, Name, VertexLabels.

Name="ReducedComplex_6_vertices_5"
Dim=3
Chi=0
F=[ 6, 14, 16, 8 ]
G=[ 1, 0 ]
H=[ 2, 2, 2, 1 ]
Homology=[ [ 0, [ ] ], [ 0, [ ] ], [ 0, [ ] ], [ 1, [ ] ] ]
IsConnected=true

/SimplicialComplex]
gap> t2:=SCCartesianProduct(SCBdSimplex(2),SCBdSimplex(2));;

gap> t2.F;

[ 9, 27, 18 ]
gap> SCBistellarOptions.WriteLevel:=0;

0
gap> SCBistellarOptions.LogLevel:=0;

0
gap> mint2:=SCReduceComplex(t2);

[ true, [SimplicialComplex

Properties known: Dim, Facets, Name, VertexLabels.

Name="unnamed complex 85"
Dim=2

/SimplicialComplex], 32 ]
```

#### 16.7 Simplicial blowups

For a more detailed description of functions related to simplicial blowups see Chapter 10.

```gap> list:=SCLib.SearchByName("Kummer");

[ [ 7493, "4-dimensional Kummer variety (VT)" ] ]

[SimplicialComplex

Properties known: AltshulerSteinberg, AutomorphismGroup,
AutomorphismGroupSize, AutomorphismGroupStructure,
AutomorphismGroupTransitivity,
ConnectedComponents, Date, Dim, DualGraph,
EulerCharacteristic, FacetsEx, GVector,
GeneratorsEx, HVector, HasBoundary, HasInterior,
Homology, Interior, IsCentrallySymmetric,
IsConnected, IsEulerianManifold, IsManifold,
IsOrientable, IsPseudoManifold, IsPure,
IsStronglyConnected, MinimalNonFacesEx, Name,
Neighborliness, NumFaces[], Orientation,
SkelExs[], Vertices.

Name="4-dimensional Kummer variety (VT)"
Dim=4
AltshulerSteinberg=45137758519296000000000000
AutomorphismGroupSize=1920
AutomorphismGroupStructure="((C2 x C2 x C2 x C2) : A5) : C2"
AutomorphismGroupTransitivity=1
EulerCharacteristic=8
GVector=[ 10, 55, 60 ]
HVector=[ 11, 66, 126, -19, 7 ]
HasBoundary=false
HasInterior=true
Homology=[ [0, [ ] ], [0, [ ] ], [6, [2,2,2,2,2] ], [0, [ ] ], [1, [ ] ] ]
IsCentrallySymmetric=false
IsConnected=true
IsEulerianManifold=true
IsOrientable=true
IsPseudoManifold=true
IsPure=true
IsStronglyConnected=true
Neighborliness=2

/SimplicialComplex]

[SimplicialComplex

Properties known: Dim, FacetsEx, Name, Vertices.

Name="lk([ 1 ]) in 4-dimensional Kummer variety (VT)"
Dim=3

/SimplicialComplex]
gap> SCHomology(lk);

[ [ 0, [  ] ], [ 0, [ 2 ] ], [ 0, [  ] ], [ 1, [  ] ] ]
gap> SCLibDetermineTopologicalType(lk);

[ 45, 113, 2426, 2502, 7470 ]

gap> d.Name;

"RP^3"
gap> SCEquivalent(lk,d);

#I  SCReduceComplexEx: complexes are bistellarly equivalent.
true
gap> e:=SCBlowup(c,1);

#I  SCBlowup: checking if singularity is a combinatorial manifold...
#I  SCBlowup: ...true
#I  SCBlowup: checking type of singularity...
#I  SCReduceComplexEx: complexes are bistellarly equivalent.
#I  SCBlowup: ...ordinary double point (supported type).
#I  SCBlowup: starting blowup...
#I  SCBlowup: map boundaries...
#I  SCBlowup: boundaries not isomorphic, initializing bistellar moves...
#I  SCBlowup: found complex with smaller boundary: f = [ 15, 74, 118, 59 ].
#I  SCBlowup: found complex with smaller boundary: f = [ 14, 70, 112, 56 ].
#I  SCBlowup: found complex with smaller boundary: f = [ 14, 69, 110, 55 ].
#I  SCBlowup: found complex with smaller boundary: f = [ 14, 68, 108, 54 ].
#I  SCBlowup: found complex with smaller boundary: f = [ 13, 64, 102, 51 ].
#I  SCBlowup: found complex with smaller boundary: f = [ 13, 63, 100, 50 ].
#I  SCBlowup: found complex with smaller boundary: f = [ 13, 62, 98, 49 ].
#I  SCBlowup: found complex with smaller boundary: f = [ 12, 58, 92, 46 ].
#I  SCBlowup: found complex with smaller boundary: f = [ 12, 57, 90, 45 ].
#I  SCBlowup: found complex with smaller boundary: f = [ 12, 56, 88, 44 ].
#I  SCBlowup: found complex with smaller boundary: f = [ 11, 52, 82, 41 ].
#I  SCBlowup: found complex with smaller boundary: f = [ 11, 51, 80, 40 ].
#I  SCBlowup: found complex with isomorphic boundaries.
#I  SCBlowup: ...boundaries mapped succesfully.
#I  SCBlowup: build complex...
#I  SCBlowup: ...done.
#I  SCBlowup: ...blowup completed.
#I  SCBlowup: You may now want to reduce the complex via 'SCReduceComplex'.
[SimplicialComplex

Properties known: Dim, FacetsEx, Name, Vertices.

Name="unnamed complex 6315 \ star([ 1 ]) in unnamed complex 6315 cup unnamed\
complex 6319 cup unnamed complex 6317"
Dim=4

/SimplicialComplex]
gap> SCHomology(c);

[ [ 0, [  ] ], [ 0, [  ] ], [ 6, [ 2, 2, 2, 2, 2 ] ], [ 0, [  ] ], [ 1, [  ] ] ]
gap> SCHomology(e);

[ [ 0, [  ] ], [ 0, [  ] ], [ 7, [ 2, 2, 2, 2 ] ], [ 0, [  ] ], [ 1, [  ] ] ]
```

#### 16.8 Discrete normal surfaces and slicings

For a more detailed description of functions related to discrete normal surfaces and slicings see the Sections 2.4 and 2.5.

```

gap> # the boundary of the cyclic 4-polytope with 6 vertices

gap> c:=SCBdCyclicPolytope(4,6);

[SimplicialComplex

Properties known: Dim, EulerCharacteristic, FacetsEx, HasBoundary, Homology,\
IsConnected, IsStronglyConnected, Name, NumFaces[], TopologicalType, Vertices.

Name="Bd(C_4(6))"
Dim=3
EulerCharacteristic=0
HasBoundary=false
Homology=[ [ 0, [ ] ], [ 0, [ ] ], [ 0, [ ] ], [ 1, [ ] ] ]
IsConnected=true
IsStronglyConnected=true
TopologicalType="S^3"

/SimplicialComplex]
gap> # slicing in between the odd and the even vertex labels, a polyhedral torus

gap> sl:=SCSlicing(c,[[2,4,6],[1,3,5]]);

[NormalSurface

Properties known: ConnectedComponents, Dim, EulerCharacteristic, FVector,\
FacetsEx, Genus, IsConnected, IsOrientable, NSTriangulation, Name,\
TopologicalType, Vertices.

Name="slicing [ [ 2, 4, 6 ], [ 1, 3, 5 ] ] of Bd(C_4(6))"
Dim=2
FVector=[ 9, 18, 0, 9 ]
EulerCharacteristic=0
IsOrientable=true
TopologicalType="T^2"

/NormalSurface]
gap> sl.Homology;

[ [ 0, [  ] ], [ 2, [  ] ], [ 1, [  ] ] ]
gap> sl.Genus;

1
gap> sl.F; # the slicing constists of 9 quadrilaterals and 0 triangles

[ 9, 18, 0, 9 ]
gap> PrintArray(sl.Facets);

[ [  [ 2, 1 ],  [ 2, 3 ],  [ 4, 1 ],  [ 4, 3 ] ],
[  [ 2, 1 ],  [ 2, 3 ],  [ 6, 1 ],  [ 6, 3 ] ],
[  [ 2, 1 ],  [ 2, 5 ],  [ 4, 1 ],  [ 4, 5 ] ],
[  [ 2, 1 ],  [ 2, 5 ],  [ 6, 1 ],  [ 6, 5 ] ],
[  [ 2, 3 ],  [ 2, 5 ],  [ 4, 3 ],  [ 4, 5 ] ],
[  [ 2, 3 ],  [ 2, 5 ],  [ 6, 3 ],  [ 6, 5 ] ],
[  [ 4, 1 ],  [ 4, 3 ],  [ 6, 1 ],  [ 6, 3 ] ],
[  [ 4, 1 ],  [ 4, 5 ],  [ 6, 1 ],  [ 6, 5 ] ],
[  [ 4, 3 ],  [ 4, 5 ],  [ 6, 3 ],  [ 6, 5 ] ] ]
```

Further example computations can be found in the slides of various talks about simpcomp, available from the simpcomp homepage (`http://www.igt.uni-stuttgart.de/LstDiffgeo/simpcomp/`), and in Appendix A of [Spr10a].

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