  
  [1X17 [33X[0;0YA demo session with [5Xsimpcomp[105X[101X[1X[133X[101X
  
  [33X[0;0YThis chapter contains the transcript of a demo session with [5Xsimpcomp[105X that is
  intended to give an insight into what things can be done with this package.[133X
  
  [33X[0;0YOf  course  this  only  scratches  the  surface of the functions provided by
  [5Xsimpcomp[105X.  See  Chapters  [14X4[114X  through  [14X15[114X  for  further functions provided by
  [5Xsimpcomp[105X.[133X
  
  
  [1X17.1 [33X[0;0YCreating a [10XSCSimplicialComplex[110X[101X[1X object[133X[101X
  
  [33X[0;0YSimplicial  complex objects can either be created from a facet list (complex
  [10Xc1[110X  below), orbit representatives together with a permutation group (complex
  [10Xc2[110X)  or  difference  cycles  (complex  [10Xc3[110X, see Section [14X6.1[114X), from a function
  generating  triangulations  of  standard  complexes (complex [10Xc4[110X, see Section
  [14X6.3[114X)  or  from  a  function  constructing  infinite series for combinatorial
  (pseudo)manifolds  (complexes  [10Xc5[110X,  [10Xc6[110X, [10Xc7[110X, see Section [14X6.4[114X and the function
  prefix  [10XSCSeries...[110X).  There  are  also  functions  creating  new simplicial
  complexes  from  old,  see  Section [14X6.6[114X, which will be described in the next
  sections.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27X#first run functionality test on simpcomp
[127X[104X
    [4X[25Xgap>[125X [27XSCRunTest();
[127X[104X
    [4X[28X+ test simpcomp package, version 2.1.6
[128X[104X
    [4X[28X+ GAP4stones: 69988
[128X[104X
    [4X[28Xtrue
[128X[104X
    [4X[25Xgap>[125X [27X#all ok
[127X[104X
    [4X[25Xgap>[125X [27Xc1:=SCFromFacets([[1,2],[2,3],[3,1]]);
[127X[104X
    [4X[28X[SimplicialComplex
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X Properties known: Dim, Facets, Name, VertexLabels.
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X Name="unnamed complex 1"
[128X[104X
    [4X[28X Dim=1
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X/SimplicialComplex]
[128X[104X
    [4X[25Xgap>[125X [27XG:=Group([(2,12,11,6,8,3)(4,7,10)(5,9),(1,11,6,4,5,3,10,8,9,7,2,12)]);
[127X[104X
    [4X[28XGroup([ (2,12,11,6,8,3)(4,7,10)(5,9), (1,11,6,4,5,3,10,8,9,7,2,12) ])
[128X[104X
    [4X[25Xgap>[125X [27XStructureDescription(G);
[127X[104X
    [4X[28X"S4 x S3"
[128X[104X
    [4X[25Xgap>[125X [27XSize(G);
[127X[104X
    [4X[28X144
[128X[104X
    [4X[25Xgap>[125X [27Xc2:=SCFromGenerators(G,[[1,2,3]]);;
[127X[104X
    [4X[25Xgap>[125X [27Xc2.IsManifold;                    
[127X[104X
    [4X[28Xtrue
[128X[104X
    [4X[25Xgap>[125X [27XSCLibDetermineTopologicalType(c2);
[127X[104X
    [4X[28X[SimplicialComplex
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X Properties known: AutomorphismGroup, AutomorphismGroupSize, 
[128X[104X
    [4X[28X                   AutomorphismGroupStructure, AutomorphismGroupTransitivity,\
[128X[104X
    [4X[28X 
[128X[104X
    [4X[28X                   Boundary, Dim, Faces, Facets, Generators, HasBoundary, 
[128X[104X
    [4X[28X                   IsManifold, IsPM, Name, TopologicalType, VertexLabels, 
[128X[104X
    [4X[28X                   Vertices.
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X Name="complex from generators under group S4 x S3"
[128X[104X
    [4X[28X Dim=2
[128X[104X
    [4X[28X AutomorphismGroupSize=144
[128X[104X
    [4X[28X AutomorphismGroupStructure="S4 x S3"
[128X[104X
    [4X[28X AutomorphismGroupTransitivity=1
[128X[104X
    [4X[28X HasBoundary=false
[128X[104X
    [4X[28X IsPM=true
[128X[104X
    [4X[28X TopologicalType="T^2"
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X/SimplicialComplex]
[128X[104X
    [4X[25Xgap>[125X [27Xc3:=SCFromDifferenceCycles([[1,1,6],[3,3,2]]);
[127X[104X
    [4X[28X[SimplicialComplex
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X Properties known: Dim, Facets, Name, VertexLabels.
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X Name="complex from diffcycles [ [ 1, 1, 6 ], [ 3, 3, 2 ] ]"
[128X[104X
    [4X[28X Dim=2
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X/SimplicialComplex]
[128X[104X
    [4X[25Xgap>[125X [27Xc4:=SCBdSimplex(2);
[127X[104X
    [4X[28X[SimplicialComplex
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X Properties known: AutomorphismGroup, AutomorphismGroupOrder, 
[128X[104X
    [4X[28X                   AutomorphismGroupStructure, AutomorphismGroupTransitivity, 
[128X[104X
    [4X[28X                   Chi, Dim, F, Facets, Generators, HasBounday, Homology, 
[128X[104X
    [4X[28X                   IsConnected, IsStronglyConnected, Name, TopologicalType, 
[128X[104X
    [4X[28X                   VertexLabels.
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X Name="S^1_3"
[128X[104X
    [4X[28X Dim=1
[128X[104X
    [4X[28X AutomorphismGroupStructure="S3"
[128X[104X
    [4X[28X AutomorphismGroupTransitivity=3
[128X[104X
    [4X[28X Chi=0
[128X[104X
    [4X[28X F=[ 3, 3 ]
[128X[104X
    [4X[28X Homology=[ [ 0, [ ] ], [ 1, [ ] ] ]
[128X[104X
    [4X[28X IsConnected=true
[128X[104X
    [4X[28X IsStronglyConnected=true
[128X[104X
    [4X[28X TopologicalType="S^1"
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X/SimplicialComplex]
[128X[104X
    [4X[25Xgap>[125X [27Xc5:=SCSeriesCSTSurface(2,16);;    
[127X[104X
    [4X[25Xgap>[125X [27XSCLibDetermineTopologicalType(c5);
[127X[104X
    [4X[28X[SimplicialComplex
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X Properties known: Boundary, Dim, Faces, Facets, HasBoundary, IsPM, Name, 
[128X[104X
    [4X[28X                   TopologicalType, VertexLabels.
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X Name="cst surface S_{(2,16)} = { (2:2:12),(6:6:4) }"
[128X[104X
    [4X[28X Dim=2
[128X[104X
    [4X[28X HasBoundary=false
[128X[104X
    [4X[28X IsPM=true
[128X[104X
    [4X[28X TopologicalType="T^2 U T^2"
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X/SimplicialComplex]
[128X[104X
    [4X[25Xgap>[125X [27Xc6:=SCSeriesD2n(22);;
[127X[104X
    [4X[25Xgap>[125X [27Xc6.Homology;
[127X[104X
    [4X[28X[ [ 0, [  ] ], [ 1, [  ] ], [ 0, [ 2 ] ], [ 0, [  ] ] ]
[128X[104X
    [4X[25Xgap>[125X [27Xc6.F;
[127X[104X
    [4X[28X[ 44, 264, 440, 220 ]
[128X[104X
    [4X[25Xgap>[125X [27XSCSeriesAGL(17);
[127X[104X
    [4X[28X[ AGL(1,17), [ [ 1, 2, 4, 8, 16 ] ] ]
[128X[104X
    [4X[25Xgap>[125X [27Xc7:=SCFromGenerators(last[1],last[2]);;
[127X[104X
    [4X[25Xgap>[125X [27Xc7.AutomorphismGroupTransitivity;
[127X[104X
    [4X[28X2
[128X[104X
  [4X[32X[104X
  
  
  [1X17.2 [33X[0;0YWorking with a [10XSCSimplicialComplex[110X[101X[1X object[133X[101X
  
  [33X[0;0YAs  described  in  Section  [14X3.1[114X  there  are two several ways of accessing an
  object  of  type [10XSCSimplicialComplex[110X. 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[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xc:=SCBdSimplex(3);; # create a simplicial complex object
[127X[104X
    [4X[25Xgap>[125X [27XSCFVector(c);
[127X[104X
    [4X[28X[ 4, 6, 4 ]
[128X[104X
    [4X[25Xgap>[125X [27XSCSkel(c,0);
[127X[104X
    [4X[28X[ [ 1 ], [ 2 ], [ 3 ], [ 4 ] ]
[128X[104X
  [4X[32X[104X
  
  [33X[0;0Yis equivalent to[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xc:=SCBdSimplex(3);; # create a simplicial complex object
[127X[104X
    [4X[25Xgap>[125X [27Xc.F;
[127X[104X
    [4X[28X[ 4, 6, 4 ]
[128X[104X
    [4X[25Xgap>[125X [27Xc.Skel(0);
[127X[104X
    [4X[28X[ [ 1 ], [ 2 ], [ 3 ], [ 4 ] ]
[128X[104X
  [4X[32X[104X
  
  
  [1X17.3 [33X[0;0YCalculating properties of a [10XSCSimplicialComplex[110X[101X[1X object[133X[101X
  
  [33X[0;0Y[5Xsimpcomp[105X  provides  a  variety  of  functions  for calculating properties of
  simplicial  complexes,  see  Section  [14X6.9[114X.  All  these  properties  are only
  calculated once and stored in the [10XSCSimplicialComplex[110X object.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xc1.F;     
[127X[104X
    [4X[28X[ 3, 3 ]
[128X[104X
    [4X[25Xgap>[125X [27Xc1.FaceLattice;
[127X[104X
    [4X[28X[ [ [ 1 ], [ 2 ], [ 3 ] ], [ [ 1, 2 ], [ 1, 3 ], [ 2, 3 ] ] ]
[128X[104X
    [4X[25Xgap>[125X [27Xc1.AutomorphismGroup;
[127X[104X
    [4X[28XS3
[128X[104X
    [4X[25Xgap>[125X [27Xc1.Generators;
[127X[104X
    [4X[28X[ [ [ 1, 2 ], 3 ] ]
[128X[104X
    [4X[25Xgap>[125X [27Xc3.Facets;
[127X[104X
    [4X[28X[ [ 1, 2, 3 ], [ 1, 2, 8 ], [ 1, 3, 6 ], [ 1, 4, 6 ], [ 1, 4, 7 ], 
[128X[104X
    [4X[28X  [ 1, 7, 8 ], [ 2, 3, 4 ], [ 2, 4, 7 ], [ 2, 5, 7 ], [ 2, 5, 8 ], 
[128X[104X
    [4X[28X  [ 3, 4, 5 ], [ 3, 5, 8 ], [ 3, 6, 8 ], [ 4, 5, 6 ], [ 5, 6, 7 ], 
[128X[104X
    [4X[28X  [ 6, 7, 8 ] ]
[128X[104X
    [4X[25Xgap>[125X [27Xc3.F;
[127X[104X
    [4X[28X[ 8, 24, 16 ]
[128X[104X
    [4X[25Xgap>[125X [27Xc3.G;
[127X[104X
    [4X[28X[ 4 ]
[128X[104X
    [4X[25Xgap>[125X [27Xc3.H;
[127X[104X
    [4X[28X[ 5, 11, -1 ]
[128X[104X
    [4X[25Xgap>[125X [27Xc3.ASDet;
[127X[104X
    [4X[28X186624
[128X[104X
    [4X[25Xgap>[125X [27Xc3.Chi;
[127X[104X
    [4X[28X0
[128X[104X
    [4X[25Xgap>[125X [27Xc3.Generators;
[127X[104X
    [4X[28X[ [ [ 1, 2, 3 ], 16 ] ]
[128X[104X
    [4X[25Xgap>[125X [27Xc3.HasBoundary;
[127X[104X
    [4X[28Xfalse
[128X[104X
    [4X[25Xgap>[125X [27Xc3.IsConnected;
[127X[104X
    [4X[28Xtrue
[128X[104X
    [4X[25Xgap>[125X [27Xc3.IsCentrallySymmetric;
[127X[104X
    [4X[28Xtrue
[128X[104X
    [4X[25Xgap>[125X [27Xc3.Vertices;
[127X[104X
    [4X[28X[ 1, 2, 3, 4, 5, 6, 7, 8 ]
[128X[104X
    [4X[25Xgap>[125X [27Xc3.ConnectedComponents;
[127X[104X
    [4X[28X[ [SimplicialComplex
[128X[104X
    [4X[28X    
[128X[104X
    [4X[28X     Properties known: Dim, Facets, Name, VertexLabels.
[128X[104X
    [4X[28X    
[128X[104X
    [4X[28X     Name="Connected component #1 of complex from diffcycles [ [ 1, 1, 6 ], [ \
[128X[104X
    [4X[28X3, 3, 2 ] ]"
[128X[104X
    [4X[28X     Dim=2
[128X[104X
    [4X[28X    
[128X[104X
    [4X[28X    /SimplicialComplex] ]
[128X[104X
    [4X[25Xgap>[125X [27Xc3.UnknownProperty;
[127X[104X
    [4X[28X#I  SCPropertyObject: unhandled property 'UnknownProperty'. Handled properties\
[128X[104X
    [4X[28X are [ "Equivalent", "IsKStackedSphere", "IsManifold", "IsMovable", "Move", 
[128X[104X
    [4X[28X  "Moves", "RMoves", "ReduceAsSubcomplex", "Reduce", "ReduceEx", "Copy", 
[128X[104X
    [4X[28X  "Recalc", "ASDet", "AutomorphismGroup", "AutomorphismGroupInternal", 
[128X[104X
    [4X[28X  "Boundary", "ConnectedComponents", "Dim", "DualGraph", "Chi", "F", 
[128X[104X
    [4X[28X  "FaceLattice", "FaceLatticeEx", "Faces", "FacesEx", "Facets", "FacetsEx", 
[128X[104X
    [4X[28X  "FpBetti", "FundamentalGroup", "G", "Generators", "GeneratorsEx", "H", 
[128X[104X
    [4X[28X  "HasBoundary", "HasInterior", "Homology", "Incidences", "IncidencesEx", 
[128X[104X
    [4X[28X  "Interior", "IsCentrallySymmetric", "IsConnected", "IsEmpty", 
[128X[104X
    [4X[28X  "IsEulerianManifold", "IsHomologySphere", "IsInKd", "IsKNeighborly", 
[128X[104X
    [4X[28X  "IsOrientable", "IsPM", "IsPure", "IsShellable", "IsStronglyConnected", 
[128X[104X
    [4X[28X  "MinimalNonFaces", "MinimalNonFacesEx", "Name", "Neighborliness", 
[128X[104X
    [4X[28X  "Orientation", "Skel", "SkelEx", "SpanningTree", 
[128X[104X
    [4X[28X  "StronglyConnectedComponents", "Vertices", "VerticesEx", 
[128X[104X
    [4X[28X  "BoundaryOperatorMatrix", "HomologyBasis", "HomologyBasisAsSimplices", 
[128X[104X
    [4X[28X  "HomologyInternal", "CoboundaryOperatorMatrix", "Cohomology", 
[128X[104X
    [4X[28X  "CohomologyBasis", "CohomologyBasisAsSimplices", "CupProduct", 
[128X[104X
    [4X[28X  "IntersectionForm", "IntersectionFormParity", 
[128X[104X
    [4X[28X  "IntersectionFormDimensionality", "Load", "Save", "ExportPolymake", 
[128X[104X
    [4X[28X  "ExportLatexTable", "ExportJavaView", "LabelMax", "LabelMin", "Labels", 
[128X[104X
    [4X[28X  "Relabel", "RelabelStandard", "RelabelTransposition", "Rename", 
[128X[104X
    [4X[28X  "SortComplex", "UnlabelFace", "AlexanderDual", "CollapseGreedy", "Cone", 
[128X[104X
    [4X[28X  "DeletedJoin", "Difference", "HandleAddition", "Intersection", 
[128X[104X
    [4X[28X  "IsIsomorphic", "IsSubcomplex", "Isomorphism", "IsomorphismEx", "Join", 
[128X[104X
    [4X[28X  "Link", "Links", "Neighbors", "NeighborsEx", "Shelling", "ShellingExt", 
[128X[104X
    [4X[28X  "Shellings", "Span", "Star", "Stars", "Suspension", "Union", 
[128X[104X
    [4X[28X  "VertexIdentification", "Wedge", "DetermineTopologicalType", "Dim", 
[128X[104X
    [4X[28X  "Facets", "VertexLabels", "Name", "Vertices", "IsConnected", 
[128X[104X
    [4X[28X  "ConnectedComponents" ].
[128X[104X
    [4X[28X
[128X[104X
    [4X[28Xfail
[128X[104X
  [4X[32X[104X
  
  
  [1X17.4 [33X[0;0YCreating new complexes from a [10XSCSimplicialComplex[110X[101X[1X object[133X[101X
  
  [33X[0;0YAs  already  mentioned,  there is the possibility to generate new objects of
  type  [10XSCSimplicialComplex[110X  from  existing ones using standard constructions.
  The  functions  used in this section are described in more detail in Section
  [14X6.6[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xd:=c3+c3;
[127X[104X
    [4X[28X[SimplicialComplex
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X Properties known: Dim, Facets, Name, VertexLabels, Vertices.
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X Name="complex from diffcycles [ [ 1, 1, 6 ], [ 3, 3, 2 ] ]#+-complex from dif\
[128X[104X
    [4X[28Xfcycles [ [ 1, 1, 6 ], [ 3, 3, 2 ] ]"
[128X[104X
    [4X[28X Dim=2
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X/SimplicialComplex]
[128X[104X
    [4X[25Xgap>[125X [27XSCRename(d,"T^2#T^2");
[127X[104X
    [4X[28Xtrue
[128X[104X
    [4X[25Xgap>[125X [27XSCLink(d,1);
[127X[104X
    [4X[28X[SimplicialComplex
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X Properties known: Dim, Facets, Name, VertexLabels.
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X Name="lk(1) in T^2#T^2"
[128X[104X
    [4X[28X Dim=1
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X/SimplicialComplex]
[128X[104X
    [4X[25Xgap>[125X [27XSCStar(d,[1,2]);
[127X[104X
    [4X[28X[SimplicialComplex
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X Properties known: Dim, Facets, Name, VertexLabels.
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X Name="star([ 1, 2 ]) in T^2#T^2"
[128X[104X
    [4X[28X Dim=2
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X/SimplicialComplex]
[128X[104X
    [4X[25Xgap>[125X [27XSCRename(c3,"T^2");
[127X[104X
    [4X[28Xtrue
[128X[104X
    [4X[25Xgap>[125X [27XSCConnectedProduct(c3,4);
[127X[104X
    [4X[28X[SimplicialComplex
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X Properties known: Dim, Facets, Name, VertexLabels, Vertices.
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X Name="T^2#+-T^2#+-T^2#+-T^2"
[128X[104X
    [4X[28X Dim=2
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X/SimplicialComplex]
[128X[104X
    [4X[25Xgap>[125X [27XSCCartesianProduct(c4,c4);
[127X[104X
    [4X[28X[SimplicialComplex
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X Properties known: Dim, Facets, Name, TopologicalType, VertexLabels.
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X Name="S^1_3xS^1_3"
[128X[104X
    [4X[28X Dim=2
[128X[104X
    [4X[28X TopologicalType="S^1xS^1"
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X/SimplicialComplex]
[128X[104X
    [4X[25Xgap>[125X [27XSCCartesianPower(c4,3);
[127X[104X
    [4X[28X[SimplicialComplex
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X Properties known: Dim, Facets, Name, TopologicalType, VertexLabels.
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X Name="(S^1_3)^3"
[128X[104X
    [4X[28X Dim=3
[128X[104X
    [4X[28X TopologicalType="(S^1)^3"
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X/SimplicialComplex]
[128X[104X
  [4X[32X[104X
  
  
  [1X17.5 [33X[0;0YHomology related calculations[133X[101X
  
  [33X[0;0Y[5Xsimpcomp[105X  relies  on  the  GAP  package  homology  [DHSW11] for its homology
  computations  but  provides  further  (co-)homology  related  functions, see
  Chapter [14X8[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs2s2:=SCCartesianProduct(SCBdSimplex(3),SCBdSimplex(3));
[127X[104X
    [4X[28X[SimplicialComplex
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X Properties known: Dim, Facets, Name, TopologicalType, VertexLabels.
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X Name="S^2_4xS^2_4"
[128X[104X
    [4X[28X Dim=4
[128X[104X
    [4X[28X TopologicalType="S^2xS^2"
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X/SimplicialComplex]
[128X[104X
    [4X[25Xgap>[125X [27XSCHomology(s2s2);
[127X[104X
    [4X[28X[ [ 0, [  ] ], [ 0, [  ] ], [ 2, [  ] ], [ 0, [  ] ], [ 1, [  ] ] ]
[128X[104X
    [4X[25Xgap>[125X [27XSCHomologyInternal(s2s2);
[127X[104X
    [4X[28X[ [ 0, [  ] ], [ 0, [  ] ], [ 2, [  ] ], [ 0, [  ] ], [ 1, [  ] ] ]
[128X[104X
    [4X[25Xgap>[125X [27XSCHomologyBasis(s2s2,2);
[127X[104X
    [4X[28X[ [ 1, [ [ 1, 70 ], [ -1, 12 ], [ 1, 2 ], [ -1, 1 ] ] ], 
[128X[104X
    [4X[28X  [ 1, [ [ 1, 143 ], [ -1, 51 ], [ 1, 29 ], [ -1, 25 ] ] ] ]
[128X[104X
    [4X[25Xgap>[125X [27XSCHomologyBasisAsSimplices(s2s2,2);
[127X[104X
    [4X[28X[ [ 1, 
[128X[104X
    [4X[28X      [ [ 1, [ 2, 3, 4 ] ], [ -1, [ 1, 3, 4 ] ], [ 1, [ 1, 2, 4 ] ], [ -1, [ 1
[128X[104X
    [4X[28X                    , 2, 3 ] ] ] ], 
[128X[104X
    [4X[28X  [ 1, [ [ 1, [ 5, 9, 13 ] ], [ -1, [ 1, 9, 13 ] ], [ 1, [ 1, 5, 13 ] ], 
[128X[104X
    [4X[28X          [ -1, [ 1, 5, 9 ] ] ] ] ]
[128X[104X
    [4X[25Xgap>[125X [27XSCCohomologyBasis(s2s2,2);
[127X[104X
    [4X[28X[ [ 1, 
[128X[104X
    [4X[28X      [ [ 1, 122 ], [ 1, 115 ], [ 1, 112 ], [ 1, 111 ], [ 1, 93 ], [ 1, 90 ], 
[128X[104X
    [4X[28X          [ 1, 89 ], [ 1, 84 ], [ 1, 83 ], [ 1, 82 ], [ 1, 46 ], [ 1, 43 ], 
[128X[104X
    [4X[28X          [ 1, 42 ], [ 1, 37 ], [ 1, 36 ], [ 1, 35 ], [ 1, 28 ], [ 1, 27 ], 
[128X[104X
    [4X[28X          [ 1, 26 ], [ 1, 25 ] ] ], 
[128X[104X
    [4X[28X  [ 1, [ [ 1, 213 ], [ 1, 201 ], [ 1, 192 ], [ 1, 189 ], [ 1, 159 ], 
[128X[104X
    [4X[28X          [ 1, 150 ], [ 1, 147 ], [ 1, 131 ], [ 1, 128 ], [ 1, 125 ], 
[128X[104X
    [4X[28X          [ 1, 67 ], [ 1, 58 ], [ 1, 55 ], [ 1, 39 ], [ 1, 36 ], [ 1, 33 ], 
[128X[104X
    [4X[28X          [ 1, 10 ], [ 1, 7 ], [ 1, 4 ], [ 1, 1 ] ] ] ]
[128X[104X
    [4X[25Xgap>[125X [27XSCCohomologyBasisAsSimplices(s2s2,2);
[127X[104X
    [4X[28X[ [ 1, [ [ 1, [ 4, 8, 12 ] ], [ 1, [ 3, 8, 12 ] ], [ 1, [ 3, 7, 12 ] ], 
[128X[104X
    [4X[28X          [ 1, [ 3, 7, 11 ] ], [ 1, [ 2, 8, 12 ] ], [ 1, [ 2, 7, 12 ] ], 
[128X[104X
    [4X[28X          [ 1, [ 2, 7, 11 ] ], [ 1, [ 2, 6, 12 ] ], [ 1, [ 2, 6, 11 ] ], 
[128X[104X
    [4X[28X          [ 1, [ 2, 6, 10 ] ], [ 1, [ 1, 8, 12 ] ], [ 1, [ 1, 7, 12 ] ], 
[128X[104X
    [4X[28X          [ 1, [ 1, 7, 11 ] ], [ 1, [ 1, 6, 12 ] ], [ 1, [ 1, 6, 11 ] ], 
[128X[104X
    [4X[28X          [ 1, [ 1, 6, 10 ] ], [ 1, [ 1, 5, 12 ] ], [ 1, [ 1, 5, 11 ] ], 
[128X[104X
    [4X[28X          [ 1, [ 1, 5, 10 ] ], [ 1, [ 1, 5, 9 ] ] ] ], 
[128X[104X
    [4X[28X  [ 1, [ [ 1, [ 13, 14, 15 ] ], [ 1, [ 9, 14, 15 ] ], [ 1, [ 9, 10, 15 ] ], 
[128X[104X
    [4X[28X          [ 1, [ 9, 10, 11 ] ], [ 1, [ 5, 14, 15 ] ], [ 1, [ 5, 10, 15 ] ], 
[128X[104X
    [4X[28X          [ 1, [ 5, 10, 11 ] ], [ 1, [ 5, 6, 15 ] ], [ 1, [ 5, 6, 11 ] ], 
[128X[104X
    [4X[28X          [ 1, [ 5, 6, 7 ] ], [ 1, [ 1, 14, 15 ] ], [ 1, [ 1, 10, 15 ] ], 
[128X[104X
    [4X[28X          [ 1, [ 1, 10, 11 ] ], [ 1, [ 1, 6, 15 ] ], [ 1, [ 1, 6, 11 ] ], 
[128X[104X
    [4X[28X          [ 1, [ 1, 6, 7 ] ], [ 1, [ 1, 2, 15 ] ], [ 1, [ 1, 2, 11 ] ], 
[128X[104X
    [4X[28X          [ 1, [ 1, 2, 7 ] ], [ 1, [ 1, 2, 3 ] ] ] ] ]
[128X[104X
    [4X[25Xgap>[125X [27XPrintArray(SCIntersectionForm(s2s2));
[127X[104X
    [4X[28X[ [  0,  1 ],
[128X[104X
    [4X[28X  [  1,  0 ] ]
[128X[104X
    [4X[25Xgap>[125X [27Xc:=s2s2+s2s2;
[127X[104X
    [4X[28X[SimplicialComplex
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X Properties known: Dim, Facets, Name, VertexLabels, Vertices.
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X Name="S^2_4xS^2_4#+-S^2_4xS^2_4"
[128X[104X
    [4X[28X Dim=4
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X/SimplicialComplex]
[128X[104X
    [4X[25Xgap>[125X [27XPrintArray(SCIntersectionForm(c));
[127X[104X
    [4X[28X[ [   0,  -1,   0,   0 ],
[128X[104X
    [4X[28X  [  -1,   0,   0,   0 ],
[128X[104X
    [4X[28X  [   0,   0,   0,  -1 ],
[128X[104X
    [4X[28X  [   0,   0,  -1,   0 ] ]
[128X[104X
  [4X[32X[104X
  
  
  [1X17.6 [33X[0;0YBistellar flips[133X[101X
  
  [33X[0;0YFor  a  more detailed description of functions related to bistellar flips as
  well as a very short introduction into the topic, see Chapter [14X9[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xbeta4:=SCBdCrossPolytope(4);;    
[127X[104X
    [4X[25Xgap>[125X [27Xs3:=SCBdSimplex(4);;             
[127X[104X
    [4X[25Xgap>[125X [27XSCEquivalent(beta4,s3);
[127X[104X
    [4X[28X#I  round 0, move: [ [ 2, 6, 7 ], [ 3, 4 ] ]
[128X[104X
    [4X[28X[ 8, 25, 34, 17 ]
[128X[104X
    [4X[28X#I  round 1, move: [ [ 2, 7 ], [ 3, 4, 5 ] ]
[128X[104X
    [4X[28X[ 8, 24, 32, 16 ]
[128X[104X
    [4X[28X#I  round 2, move: [ [ 2, 5 ], [ 3, 4, 8 ] ]
[128X[104X
    [4X[28X[ 8, 23, 30, 15 ]
[128X[104X
    [4X[28X#I  round 3, move: [ [ 2 ], [ 3, 4, 6, 8 ] ]
[128X[104X
    [4X[28X[ 7, 19, 24, 12 ]
[128X[104X
    [4X[28X#I  round 4, move: [ [ 6, 8 ], [ 1, 3, 4 ] ]
[128X[104X
    [4X[28X[ 7, 18, 22, 11 ]
[128X[104X
    [4X[28X#I  round 5, move: [ [ 8 ], [ 1, 3, 4, 5 ] ]
[128X[104X
    [4X[28X[ 6, 14, 16, 8 ]
[128X[104X
    [4X[28X#I  round 6, move: [ [ 5 ], [ 1, 3, 4, 7 ] ]
[128X[104X
    [4X[28X[ 5, 10, 10, 5 ]
[128X[104X
    [4X[28X#I  SCReduceComplexEx: complexes are bistellarly equivalent.
[128X[104X
    [4X[28Xtrue
[128X[104X
    [4X[25Xgap>[125X [27XSCBistellarOptions.WriteLevel;   
[127X[104X
    [4X[28X0
[128X[104X
    [4X[25Xgap>[125X [27XSCBistellarOptions.WriteLevel:=1;
[127X[104X
    [4X[28X1
[128X[104X
    [4X[25Xgap>[125X [27XSCEquivalent(beta4,s3);          
[127X[104X
    [4X[28X#I  SCLibInit: made directory "~/PATH" for user library.
[128X[104X
    [4X[28X#I  SCIntFunc.SCLibInit: index not found -- trying to reconstruct it.
[128X[104X
    [4X[28X#I  SCLibUpdate: rebuilding index for ~/PATH.
[128X[104X
    [4X[28X#I  SCLibUpdate: rebuilding index done.
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X#I  round 0, move: [ [ 2, 4, 6 ], [ 7, 8 ] ]
[128X[104X
    [4X[28X[ 8, 25, 34, 17 ]
[128X[104X
    [4X[28X#I  round 1, move: [ [ 2, 4 ], [ 5, 7, 8 ] ]
[128X[104X
    [4X[28X[ 8, 24, 32, 16 ]
[128X[104X
    [4X[28X#I  round 2, move: [ [ 4, 5 ], [ 1, 7, 8 ] ]
[128X[104X
    [4X[28X[ 8, 23, 30, 15 ]
[128X[104X
    [4X[28X#I  round 3, move: [ [ 4 ], [ 1, 6, 7, 8 ] ]
[128X[104X
    [4X[28X[ 7, 19, 24, 12 ]
[128X[104X
    [4X[28X#I  SCLibAdd: saving complex to file "complex_ReducedComplex_7_vertices_3_2009\
[128X[104X
    [4X[28X-10-27_11-40-00.sc".
[128X[104X
    [4X[28X#I  round 4, move: [ [ 2, 6 ], [ 3, 7, 8 ] ]
[128X[104X
    [4X[28X[ 7, 18, 22, 11 ]
[128X[104X
    [4X[28X#I  round 5, move: [ [ 2 ], [ 3, 5, 7, 8 ] ]
[128X[104X
    [4X[28X[ 6, 14, 16, 8 ]
[128X[104X
    [4X[28X#I  SCLibAdd: saving complex to file "complex_ReducedComplex_6_vertices_5_2009\
[128X[104X
    [4X[28X-10-27_11-40-00.sc".
[128X[104X
    [4X[28X#I  round 6, move: [ [ 5 ], [ 1, 3, 7, 8 ] ]
[128X[104X
    [4X[28X[ 5, 10, 10, 5 ]
[128X[104X
    [4X[28X#I  SCLibAdd: saving complex to file "complex_ReducedComplex_5_vertices_6_2009\
[128X[104X
    [4X[28X-10-27_11-40-00.sc".
[128X[104X
    [4X[28X#I  SCLibAdd: saving complex to file "complex_ReducedComplex_5_vertices_7_2009\
[128X[104X
    [4X[28X-10-27_11-40-00.sc".
[128X[104X
    [4X[28X#I  SCReduceComplexEx: complexes are bistellarly equivalent.
[128X[104X
    [4X[28Xtrue
[128X[104X
    [4X[25Xgap>[125X [27XmyLib:=SCLibInit("~/PATH"); # copy path from above             
[127X[104X
    [4X[28X[Simplicial complex library. Properties:
[128X[104X
    [4X[28XCalculateIndexAttributes=true
[128X[104X
    [4X[28XNumber of complexes in library=4
[128X[104X
    [4X[28XIndexAttributes=[ "Name", "Date", "Dim", "F", "G", "H", "Chi", "Homology" ]
[128X[104X
    [4X[28XLoaded=true
[128X[104X
    [4X[28XPath="/home/spreerjn/reducedComplexes/2009-10-27_11-40-00/"
[128X[104X
    [4X[28X]
[128X[104X
    [4X[25Xgap>[125X [27Xs3:=myLib.Load(3);
[127X[104X
    [4X[28X[SimplicialComplex
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X Properties known: Chi, Date, Dim, F, Faces, Facets, G, H, Homology, 
[128X[104X
    [4X[28X                   IsConnected, Name, VertexLabels.
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X Name="ReducedComplex_5_vertices_6"
[128X[104X
    [4X[28X Dim=3
[128X[104X
    [4X[28X Chi=0
[128X[104X
    [4X[28X F=[ 5, 10, 10, 5 ]
[128X[104X
    [4X[28X G=[ 0, 0 ]
[128X[104X
    [4X[28X H=[ 1, 1, 1, 1 ]
[128X[104X
    [4X[28X Homology=[ [ 0, [ ] ], [ 0, [ ] ], [ 0, [ ] ], [ 1, [ ] ] ]
[128X[104X
    [4X[28X IsConnected=true
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X/SimplicialComplex]
[128X[104X
    [4X[25Xgap>[125X [27Xs3:=myLib.Load(2);
[127X[104X
    [4X[28X[SimplicialComplex
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X Properties known: Chi, Date, Dim, F, Faces, Facets, G, H, Homology, 
[128X[104X
    [4X[28X                   IsConnected, Name, VertexLabels.
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X Name="ReducedComplex_6_vertices_5"
[128X[104X
    [4X[28X Dim=3
[128X[104X
    [4X[28X Chi=0
[128X[104X
    [4X[28X F=[ 6, 14, 16, 8 ]
[128X[104X
    [4X[28X G=[ 1, 0 ]
[128X[104X
    [4X[28X H=[ 2, 2, 2, 1 ]
[128X[104X
    [4X[28X Homology=[ [ 0, [ ] ], [ 0, [ ] ], [ 0, [ ] ], [ 1, [ ] ] ]
[128X[104X
    [4X[28X IsConnected=true
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X/SimplicialComplex]
[128X[104X
    [4X[25Xgap>[125X [27Xt2:=SCCartesianProduct(SCBdSimplex(2),SCBdSimplex(2));;
[127X[104X
    [4X[25Xgap>[125X [27Xt2.F;
[127X[104X
    [4X[28X[ 9, 27, 18 ]
[128X[104X
    [4X[25Xgap>[125X [27XSCBistellarOptions.WriteLevel:=0;
[127X[104X
    [4X[28X0
[128X[104X
    [4X[25Xgap>[125X [27XSCBistellarOptions.LogLevel:=0;  
[127X[104X
    [4X[28X0
[128X[104X
    [4X[25Xgap>[125X [27Xmint2:=SCReduceComplex(t2);    
[127X[104X
    [4X[28X[ true, [SimplicialComplex
[128X[104X
    [4X[28X    
[128X[104X
    [4X[28X     Properties known: Dim, Facets, Name, VertexLabels.
[128X[104X
    [4X[28X    
[128X[104X
    [4X[28X     Name="unnamed complex 85"
[128X[104X
    [4X[28X     Dim=2
[128X[104X
    [4X[28X    
[128X[104X
    [4X[28X    /SimplicialComplex], 32 ]
[128X[104X
    [4X[28X			[128X[104X
  [4X[32X[104X
  
  
  [1X17.7 [33X[0;0YSimplicial blowups[133X[101X
  
  [33X[0;0YFor  a  more detailed description of functions related to simplicial blowups
  see Chapter [14X10[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xlist:=SCLib.SearchByName("Kummer");
[127X[104X
    [4X[28X[ [ 7493, "4-dimensional Kummer variety (VT)" ] ]
[128X[104X
    [4X[25Xgap>[125X [27Xc:=SCLib.Load(7493);
[127X[104X
    [4X[28X[SimplicialComplex
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X Properties known: AltshulerSteinberg, AutomorphismGroup, 
[128X[104X
    [4X[28X                   AutomorphismGroupSize, AutomorphismGroupStructure, 
[128X[104X
    [4X[28X                   AutomorphismGroupTransitivity, 
[128X[104X
    [4X[28X                   ConnectedComponents, Date, Dim, DualGraph, 
[128X[104X
    [4X[28X                   EulerCharacteristic, FacetsEx, GVector, 
[128X[104X
    [4X[28X                   GeneratorsEx, HVector, HasBoundary, HasInterior, 
[128X[104X
    [4X[28X                   Homology, Interior, IsCentrallySymmetric, 
[128X[104X
    [4X[28X                   IsConnected, IsEulerianManifold, IsManifold, 
[128X[104X
    [4X[28X                   IsOrientable, IsPseudoManifold, IsPure, 
[128X[104X
    [4X[28X                   IsStronglyConnected, MinimalNonFacesEx, Name, 
[128X[104X
    [4X[28X                   Neighborliness, NumFaces[], Orientation, 
[128X[104X
    [4X[28X                   SkelExs[], Vertices.
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X Name="4-dimensional Kummer variety (VT)"
[128X[104X
    [4X[28X Dim=4
[128X[104X
    [4X[28X AltshulerSteinberg=45137758519296000000000000
[128X[104X
    [4X[28X AutomorphismGroupSize=1920
[128X[104X
    [4X[28X AutomorphismGroupStructure="((C2 x C2 x C2 x C2) : A5) : C2"
[128X[104X
    [4X[28X AutomorphismGroupTransitivity=1
[128X[104X
    [4X[28X EulerCharacteristic=8
[128X[104X
    [4X[28X GVector=[ 10, 55, 60 ]
[128X[104X
    [4X[28X HVector=[ 11, 66, 126, -19, 7 ]
[128X[104X
    [4X[28X HasBoundary=false
[128X[104X
    [4X[28X HasInterior=true
[128X[104X
    [4X[28X Homology=[ [0, [ ] ], [0, [ ] ], [6, [2,2,2,2,2] ], [0, [ ] ], [1, [ ] ] ]
[128X[104X
    [4X[28X IsCentrallySymmetric=false
[128X[104X
    [4X[28X IsConnected=true
[128X[104X
    [4X[28X IsEulerianManifold=true
[128X[104X
    [4X[28X IsOrientable=true
[128X[104X
    [4X[28X IsPseudoManifold=true
[128X[104X
    [4X[28X IsPure=true
[128X[104X
    [4X[28X IsStronglyConnected=true
[128X[104X
    [4X[28X Neighborliness=2
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X/SimplicialComplex]
[128X[104X
    [4X[25Xgap>[125X [27Xlk:=SCLink(c,1);
[127X[104X
    [4X[28X[SimplicialComplex
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X Properties known: Dim, FacetsEx, Name, Vertices.
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X Name="lk([ 1 ]) in 4-dimensional Kummer variety (VT)"
[128X[104X
    [4X[28X Dim=3
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X/SimplicialComplex]
[128X[104X
    [4X[25Xgap>[125X [27XSCHomology(lk);
[127X[104X
    [4X[28X[ [ 0, [  ] ], [ 0, [ 2 ] ], [ 0, [  ] ], [ 1, [  ] ] ]
[128X[104X
    [4X[25Xgap>[125X [27XSCLibDetermineTopologicalType(lk);
[127X[104X
    [4X[28X[ 45, 113, 2426, 2502, 7470 ]
[128X[104X
    [4X[25Xgap>[125X [27Xd:=SCLib.Load(45);;
[127X[104X
    [4X[25Xgap>[125X [27Xd.Name;
[127X[104X
    [4X[28X"RP^3"
[128X[104X
    [4X[25Xgap>[125X [27XSCEquivalent(lk,d);
[127X[104X
    [4X[28X#I  SCReduceComplexEx: complexes are bistellarly equivalent.
[128X[104X
    [4X[28Xtrue
[128X[104X
    [4X[25Xgap>[125X [27Xe:=SCBlowup(c,1);
[127X[104X
    [4X[28X#I  SCBlowup: checking if singularity is a combinatorial manifold...
[128X[104X
    [4X[28X#I  SCBlowup: ...true
[128X[104X
    [4X[28X#I  SCBlowup: checking type of singularity...
[128X[104X
    [4X[28X#I  SCReduceComplexEx: complexes are bistellarly equivalent.
[128X[104X
    [4X[28X#I  SCBlowup: ...ordinary double point (supported type).
[128X[104X
    [4X[28X#I  SCBlowup: starting blowup...
[128X[104X
    [4X[28X#I  SCBlowup: map boundaries...
[128X[104X
    [4X[28X#I  SCBlowup: boundaries not isomorphic, initializing bistellar moves...
[128X[104X
    [4X[28X#I  SCBlowup: found complex with smaller boundary: f = [ 15, 74, 118, 59 ].
[128X[104X
    [4X[28X#I  SCBlowup: found complex with smaller boundary: f = [ 14, 70, 112, 56 ].
[128X[104X
    [4X[28X#I  SCBlowup: found complex with smaller boundary: f = [ 14, 69, 110, 55 ].
[128X[104X
    [4X[28X#I  SCBlowup: found complex with smaller boundary: f = [ 14, 68, 108, 54 ].
[128X[104X
    [4X[28X#I  SCBlowup: found complex with smaller boundary: f = [ 13, 64, 102, 51 ].
[128X[104X
    [4X[28X#I  SCBlowup: found complex with smaller boundary: f = [ 13, 63, 100, 50 ].
[128X[104X
    [4X[28X#I  SCBlowup: found complex with smaller boundary: f = [ 13, 62, 98, 49 ].
[128X[104X
    [4X[28X#I  SCBlowup: found complex with smaller boundary: f = [ 12, 58, 92, 46 ].
[128X[104X
    [4X[28X#I  SCBlowup: found complex with smaller boundary: f = [ 12, 57, 90, 45 ].
[128X[104X
    [4X[28X#I  SCBlowup: found complex with smaller boundary: f = [ 12, 56, 88, 44 ].
[128X[104X
    [4X[28X#I  SCBlowup: found complex with smaller boundary: f = [ 11, 52, 82, 41 ].
[128X[104X
    [4X[28X#I  SCBlowup: found complex with smaller boundary: f = [ 11, 51, 80, 40 ].
[128X[104X
    [4X[28X#I  SCBlowup: found complex with isomorphic boundaries.
[128X[104X
    [4X[28X#I  SCBlowup: ...boundaries mapped succesfully.
[128X[104X
    [4X[28X#I  SCBlowup: build complex...
[128X[104X
    [4X[28X#I  SCBlowup: ...done.
[128X[104X
    [4X[28X#I  SCBlowup: ...blowup completed.
[128X[104X
    [4X[28X#I  SCBlowup: You may now want to reduce the complex via 'SCReduceComplex'.
[128X[104X
    [4X[28X[SimplicialComplex
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X Properties known: Dim, FacetsEx, Name, Vertices.
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X Name="unnamed complex 6315 \ star([ 1 ]) in unnamed complex 6315 cup unnamed\
[128X[104X
    [4X[28X complex 6319 cup unnamed complex 6317"
[128X[104X
    [4X[28X Dim=4
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X/SimplicialComplex]
[128X[104X
    [4X[25Xgap>[125X [27XSCHomology(c);
[127X[104X
    [4X[28X[ [ 0, [  ] ], [ 0, [  ] ], [ 6, [ 2, 2, 2, 2, 2 ] ], [ 0, [  ] ], [ 1, [  ] ] ]
[128X[104X
    [4X[25Xgap>[125X [27XSCHomology(e);
[127X[104X
    [4X[28X[ [ 0, [  ] ], [ 0, [  ] ], [ 7, [ 2, 2, 2, 2 ] ], [ 0, [  ] ], [ 1, [  ] ] ]
[128X[104X
    [4X[28X			[128X[104X
  [4X[32X[104X
  
  
  [1X17.8 [33X[0;0YDiscrete normal surfaces and slicings[133X[101X
  
  [33X[0;0YFor  a  more  detailed  description  of functions related to discrete normal
  surfaces and slicings see the Sections [14X2.4[114X and [14X2.5[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X		
[128X[104X
    [4X[25Xgap>[125X [27X# the boundary of the cyclic 4-polytope with 6 vertices		
[127X[104X
    [4X[25Xgap>[125X [27Xc:=SCBdCyclicPolytope(4,6); 
[127X[104X
    [4X[28X[SimplicialComplex
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X Properties known: Dim, EulerCharacteristic, FacetsEx, HasBoundary, Homology,\
[128X[104X
    [4X[28X IsConnected, IsStronglyConnected, Name, NumFaces[], TopologicalType, Vertices.
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X Name="Bd(C_4(6))"
[128X[104X
    [4X[28X Dim=3
[128X[104X
    [4X[28X EulerCharacteristic=0
[128X[104X
    [4X[28X HasBoundary=false
[128X[104X
    [4X[28X Homology=[ [ 0, [ ] ], [ 0, [ ] ], [ 0, [ ] ], [ 1, [ ] ] ]
[128X[104X
    [4X[28X IsConnected=true
[128X[104X
    [4X[28X IsStronglyConnected=true
[128X[104X
    [4X[28X TopologicalType="S^3"
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X/SimplicialComplex]
[128X[104X
    [4X[25Xgap>[125X [27X# slicing in between the odd and the even vertex labels, a polyhedral torus
[127X[104X
    [4X[25Xgap>[125X [27Xsl:=SCSlicing(c,[[2,4,6],[1,3,5]]);   
[127X[104X
    [4X[28X[NormalSurface
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X Properties known: ConnectedComponents, Dim, EulerCharacteristic, FVector,\
[128X[104X
    [4X[28X FacetsEx, Genus, IsConnected, IsOrientable, NSTriangulation, Name,\
[128X[104X
    [4X[28X TopologicalType, Vertices.
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X Name="slicing [ [ 2, 4, 6 ], [ 1, 3, 5 ] ] of Bd(C_4(6))"
[128X[104X
    [4X[28X Dim=2
[128X[104X
    [4X[28X FVector=[ 9, 18, 0, 9 ]
[128X[104X
    [4X[28X EulerCharacteristic=0
[128X[104X
    [4X[28X IsOrientable=true
[128X[104X
    [4X[28X TopologicalType="T^2"
[128X[104X
    [4X[28X
[128X[104X
    [4X[28X/NormalSurface]
[128X[104X
    [4X[25Xgap>[125X [27Xsl.Homology;
[127X[104X
    [4X[28X[ [ 0, [  ] ], [ 2, [  ] ], [ 1, [  ] ] ]
[128X[104X
    [4X[25Xgap>[125X [27Xsl.Genus;
[127X[104X
    [4X[28X1
[128X[104X
    [4X[25Xgap>[125X [27Xsl.F; # the slicing constists of 9 quadrilaterals and 0 triangles
[127X[104X
    [4X[28X[ 9, 18, 0, 9 ]
[128X[104X
    [4X[25Xgap>[125X [27XPrintArray(sl.Facets);
[127X[104X
    [4X[28X[ [  [ 2, 1 ],  [ 2, 3 ],  [ 4, 1 ],  [ 4, 3 ] ],
[128X[104X
    [4X[28X  [  [ 2, 1 ],  [ 2, 3 ],  [ 6, 1 ],  [ 6, 3 ] ],
[128X[104X
    [4X[28X  [  [ 2, 1 ],  [ 2, 5 ],  [ 4, 1 ],  [ 4, 5 ] ],
[128X[104X
    [4X[28X  [  [ 2, 1 ],  [ 2, 5 ],  [ 6, 1 ],  [ 6, 5 ] ],
[128X[104X
    [4X[28X  [  [ 2, 3 ],  [ 2, 5 ],  [ 4, 3 ],  [ 4, 5 ] ],
[128X[104X
    [4X[28X  [  [ 2, 3 ],  [ 2, 5 ],  [ 6, 3 ],  [ 6, 5 ] ],
[128X[104X
    [4X[28X  [  [ 4, 1 ],  [ 4, 3 ],  [ 6, 1 ],  [ 6, 3 ] ],
[128X[104X
    [4X[28X  [  [ 4, 1 ],  [ 4, 5 ],  [ 6, 1 ],  [ 6, 5 ] ],
[128X[104X
    [4X[28X  [  [ 4, 3 ],  [ 4, 5 ],  [ 6, 3 ],  [ 6, 5 ] ] ]
[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFurther  example  computations  can  be found in the slides of various talks
  about     [5Xsimpcomp[105X,     available     from     the     [5Xsimpcomp[105X     homepage
  ([10Xhttps://github.com/simpcomp-team/simpcomp[110X), and in Appendix A of [Spr11a].[133X
  
