Quelle demo.xml Sprache: XML

<Chapter Label="chap:demo"> <Heading>A demo session with <Package>simpcomp</Package></Heading>
  This chapter contains the transcript of a demo session with <Package>simpcomp</Package> that is intended to give an insight into what things can be done with this package.<P/>

  Of course this only scratches the surface of the functions provided by <Package>simpcomp</Package>. See Chapters <Ref Chap="chap:polyhedralcomplex"/> through <Ref Chap="chap:misc"/> for further functions provided by <Package>simpcomp</Package>.

  <Section Label="sec:DemoCreatingCompl">
   <Heading>Creating a <C>SCSimplicialComplex</C> object</Heading>

Simplicial complex objects can either be created from a facet list (complex <C>c1</C> below), orbit representatives together with a permutation group (complex <C>c2</C>) or difference cycles (complex <C>c3</C>, see Section <Ref Chap="sec:FromScratch"/>), from a function generating triangulations of standard complexes (complex <C>c4</C>, see Section <Ref Chap="sec:Standard"/>) or from a function constructing infinite series for combinatorial (pseudo)manifolds (complexes <C>c5</C>, <C>c6</C>, <C>c7</C>, see Section <Ref Chap="sec:Series"/> and the function prefix <C>SCSeries...</C>). There are also functions creating new simplicial complexes from old, see Section <Ref Chap="sec:generateFromOld" />, which will be described in the next sections.

<Log>
gap> #first run functionality test on simpcomp
gap> SCRunTest();
+ test simpcomp package, version 0.0.0
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
</Log>
  </Section>

  <Section Label="sec:DemoWorkingCompl">
   <Heading>Working with a <C>SCSimplicialComplex</C> object</Heading>
As described in Section <Ref Sect="sec:AcessSC" /> there are two several ways of accessing an object of type <C>SCSimplicialComplex</C>. 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

<Log>
gap> c:=SCBdSimplex(3);; # create a simplicial complex object
gap> SCFVector(c);
[ 4, 6, 4 ]
gap> SCSkel(c,0);
[ [ 1 ], [ 2 ], [ 3 ], [ 4 ] ]
</Log>

is equivalent to

<Log>
gap> c:=SCBdSimplex(3);; # create a simplicial complex object
gap> c.F;
[ 4, 6, 4 ]
gap> c.Skel(0);
[ [ 1 ], [ 2 ], [ 3 ], [ 4 ] ]
</Log>

</Section>

  <Section Label="sec:DemoPropsCompl">
   <Heading>Calculating properties of a <C>SCSimplicialComplex</C> object</Heading>

   <Package>simpcomp</Package> provides a variety of functions for calculating properties of simplicial complexes, see Section <Ref Chap="sec:glprops" />. All these properties are only calculated once and stored in the <C>SCSimplicialComplex</C> object.
<Log>
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
</Log>

  </Section>

  <Section Label="sec:DemoNewCompl">
   <Heading>Creating new complexes from a <C>SCSimplicialComplex</C> object</Heading>

   As already mentioned, there is the possibility to generate new objects of type <C>SCSimplicialComplex</C> from existing ones using standard constructions. The functions used in this section are described in more detail in Section <Ref Chap="sec:generateFromOld"/>.
<Log>
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]
</Log>

  </Section>

  <Section Label="sec:DemoHom">
   <Heading>Homology related calculations</Heading>

   <Package>simpcomp</Package> relies on the GAP package homology <Cite Key="Dumas04Homology" /> for its homology computations but provides further (co-)homology related functions, see Chapter <Ref Chap="chap:homology"/>.


<Log>
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 ] ]
</Log>

   </Section>

  <Section Label="sec:DemoBist">
   <Heading>Bistellar flips</Heading>

   For a more detailed description of functions related to bistellar flips as well as a very short introduction into the topic, see Chapter <Ref Chap="chap:bistellar"/>.
   <Log>
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 ]
   </Log>
  </Section>

  <Section Label="sec:DemoBlowups">
   <Heading>Simplicial blowups</Heading>

   For a more detailed description of functions related to simplicial blowups see Chapter <Ref Chap="chap:blowups"/>.
   <Log>
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, [  ] ] ]
   </Log>
  </Section>


  <Section Label="sec:DemoDiscreteNormalSurface">
   <Heading>Discrete normal surfaces and slicings</Heading>

   For a more detailed description of functions related to discrete normal surfaces and slicings see the Sections <Ref Chap="sec:NormSurfTheory"/> and <Ref Chap="sec:MorseTheory"/>.
   <Log>

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 ] ] ]
</Log></Section>

Further example computations can be found in the slides of various talks about <Package>simpcomp</Package>, available from the <Package>simpcomp</Package> homepage (<C>https://github.com/simpcomp-team/simpcomp</C>), and in Appendix A of <Cite Key="Spreer10Diss"/>.
</Chapter>

quality93%

¤ Dauer der Verarbeitung: 0.15 Sekunden (vorverarbeitet) ¤

Wurzel

Suchen

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung ist noch experimentell.