chap17.txt

  
  17 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  15  for  further functions provided by
  simpcomp.
  
  
  17.1 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
  6.3)  or  from  a  function  constructing  infinite series for combinatorial
  (pseudo)manifolds  (complexes  c5,  c6, c7, see Section 6.4 and the function
  prefix  SCSeries...).  There  are  also  functions  creating  new simplicial
  complexes  from  old,  see  Section 6.6, which will be described in the next
  sections.
  
    Example  
    gap> #first run functionality test on simpcomp
    gap> SCRunTest();
    + test simpcomp package, version 2.1.7
    + 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
  
  
  
  17.2 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
  
    Example  
    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
  
    Example  
    gap> c:=SCBdSimplex(3);; # create a simplicial complex object
    gap> c.F;
    [ 4, 6, 4 ]
    gap> c.Skel(0);
    [ [ 1 ], [ 2 ], [ 3 ], [ 4 ] ]
  
  
  
  17.3 Calculating properties of a SCSimplicialComplex object
  
  simpcomp  provides  a  variety  of  functions  for calculating properties of
  simplicial  complexes,  see  Section  6.9.  All  these  properties  are only
  calculated once and stored in the SCSimplicialComplex object.
  
    Example  
    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", 
      "IntersectionFormDimensionality", "Load", "Save", "ExportPolymake", 
      "ExportLatexTable", "ExportJavaView", "LabelMax", "LabelMin", "Labels", 
      "Relabel", "RelabelStandard", "RelabelTransposition", "Rename", 
      "SortComplex", "UnlabelFace", "AlexanderDual", "CollapseGreedy", "Cone", 
      "DeletedJoin", "Difference", "HandleAddition", "Intersection", 
      "IsIsomorphic", "IsSubcomplex", "Isomorphism", "IsomorphismEx", "Join", 
      "Link", "Links", "Neighbors", "NeighborsEx", "Shelling", "ShellingExt", 
      "Shellings", "Span", "Star", "Stars", "Suspension", "Union", 
      "VertexIdentification", "Wedge", "DetermineTopologicalType", "Dim", 
      "Facets", "VertexLabels", "Name", "Vertices", "IsConnected", 
      "ConnectedComponents" ].
    
    fail
  
  
  
  17.4 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
  6.6.
  
    Example  
    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
    gap> SCLink(d,1);
    [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]
  
  
  
  17.5 Homology related calculations
  
  simpcomp  relies  on  the  GAP  package  homology  [DHSW11] for its homology
  computations  but  provides  further  (co-)homology  related  functions, see
  Chapter 8.
  
    Example  
    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 ] ]
  
  
  
  17.6 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.
  
    Example  
    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  SCIntFunc.SCLibInit: index not found -- trying to reconstruct it.
    #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" ]
    Loaded=true
    Path="/home/spreerjn/reducedComplexes/2009-10-27_11-40-00/"
    ]
    gap> s3:=myLib.Load(3);
    [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]
    gap> s3:=myLib.Load(2);
    [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 ]
    			
  
  
  
  17.7 Simplicial blowups
  
  For  a  more detailed description of functions related to simplicial blowups
  see Chapter 10.
  
    Example  
    gap> list:=SCLib.SearchByName("Kummer");
    [ [ 7493, "4-dimensional Kummer variety (VT)" ] ]
    gap> c:=SCLib.Load(7493);
    [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]
    gap> lk:=SCLink(c,1);
    [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:=SCLib.Load(45);;
    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, [  ] ] ]
    			
  
  
  
  17.8 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.
  
    Example  
    		
    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
  (https://github.com/simpcomp-team/simpcomp), and in Appendix A of [Spr11a].