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 through for further functions provided by SimpComp.

Creating a SCSimplicialComplex object

Simplicial complex objects can either be created from a facet list, from a function generating triangulations of standard complexes or by expanding difference cycles. There are also functions creating new simplicial complexes from old, see chapter which will be described in the next sections.

gap> #first run functionality test on simpcomp gap> SCRunTest(); + test simpcomp package, version 1.1.21 + 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> c2:=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> 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>
Working with a SCSimplicialComplex object
As described in section 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 ] ] gap>

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 ] ] gap>
Calculating properties of a SCSimplicialComplex object

SimpComp provides a variety of functions for calculating properties of simplicial complexes, see chapter . All these properties are only calculated once and stored in the simplicial complex 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", "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 gap>

Creating new complexes from a SCSimplicialComplex object
gap> c4:=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(c4,"T^2#T^2"); true gap> SCLink(c4,1); [SimplicialComplex

Properties known: Dim, Facets, Name, VertexLabels.

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

/SimplicialComplex] gap> SCStar(c4,[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(c2,c2); [SimplicialComplex

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

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

/SimplicialComplex] gap> SCCartesianPower(c2,3); [SimplicialComplex

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

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

/SimplicialComplex] gap>
Homology related calculations

SimpComp relies on the GAP package homology for its homology computations but provides further (co-)homology related functions, see chapter .

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 ] ] gap>
Bistellar flips
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 ] gap>