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.
SCSimplicialComplex objectSimplicial 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.
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
SCSimplicialComplex objectAs described in Section 3.1 there are two several ways of accessing an object of type SCSimplicialComplex. An example for the two equivalent ways is given below. The preference will be given to the object oriented notation in this demo session. The code listed below
gap> c:=SCBdSimplex(3);; # create a simplicial complex object gap> SCFVector(c); [ 4, 6, 4 ] gap> SCSkel(c,0); [ [ 1 ], [ 2 ], [ 3 ], [ 4 ] ]
is equivalent to
gap> c:=SCBdSimplex(3);; # create a simplicial complex object gap> c.F; [ 4, 6, 4 ] gap> c.Skel(0); [ [ 1 ], [ 2 ], [ 3 ], [ 4 ] ]
SCSimplicialComplex objectsimpcomp 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.
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
SCSimplicialComplex objectAs 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.
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]
simpcomp relies on the GAP package homology [DHSW11] for its homology computations but provides further (co-)homology related functions, see Chapter 8.
gap> s2s2:=SCCartesianProduct(SCBdSimplex(3),SCBdSimplex(3)); [SimplicialComplex Properties known: Dim, Facets, Name, TopologicalType, VertexLabels. Name="S^2_4xS^2_4" Dim=4 TopologicalType="S^2xS^2" /SimplicialComplex] gap> SCHomology(s2s2); [ [ 0, [ ] ], [ 0, [ ] ], [ 2, [ ] ], [ 0, [ ] ], [ 1, [ ] ] ] gap> SCHomologyInternal(s2s2); [ [ 0, [ ] ], [ 0, [ ] ], [ 2, [ ] ], [ 0, [ ] ], [ 1, [ ] ] ] gap> SCHomologyBasis(s2s2,2); [ [ 1, [ [ 1, 70 ], [ -1, 12 ], [ 1, 2 ], [ -1, 1 ] ] ], [ 1, [ [ 1, 143 ], [ -1, 51 ], [ 1, 29 ], [ -1, 25 ] ] ] ] gap> SCHomologyBasisAsSimplices(s2s2,2); [ [ 1, [ [ 1, [ 2, 3, 4 ] ], [ -1, [ 1, 3, 4 ] ], [ 1, [ 1, 2, 4 ] ], [ -1, [ 1 , 2, 3 ] ] ] ], [ 1, [ [ 1, [ 5, 9, 13 ] ], [ -1, [ 1, 9, 13 ] ], [ 1, [ 1, 5, 13 ] ], [ -1, [ 1, 5, 9 ] ] ] ] ] gap> SCCohomologyBasis(s2s2,2); [ [ 1, [ [ 1, 122 ], [ 1, 115 ], [ 1, 112 ], [ 1, 111 ], [ 1, 93 ], [ 1, 90 ], [ 1, 89 ], [ 1, 84 ], [ 1, 83 ], [ 1, 82 ], [ 1, 46 ], [ 1, 43 ], [ 1, 42 ], [ 1, 37 ], [ 1, 36 ], [ 1, 35 ], [ 1, 28 ], [ 1, 27 ], [ 1, 26 ], [ 1, 25 ] ] ], [ 1, [ [ 1, 213 ], [ 1, 201 ], [ 1, 192 ], [ 1, 189 ], [ 1, 159 ], [ 1, 150 ], [ 1, 147 ], [ 1, 131 ], [ 1, 128 ], [ 1, 125 ], [ 1, 67 ], [ 1, 58 ], [ 1, 55 ], [ 1, 39 ], [ 1, 36 ], [ 1, 33 ], [ 1, 10 ], [ 1, 7 ], [ 1, 4 ], [ 1, 1 ] ] ] ] gap> SCCohomologyBasisAsSimplices(s2s2,2); [ [ 1, [ [ 1, [ 4, 8, 12 ] ], [ 1, [ 3, 8, 12 ] ], [ 1, [ 3, 7, 12 ] ], [ 1, [ 3, 7, 11 ] ], [ 1, [ 2, 8, 12 ] ], [ 1, [ 2, 7, 12 ] ], [ 1, [ 2, 7, 11 ] ], [ 1, [ 2, 6, 12 ] ], [ 1, [ 2, 6, 11 ] ], [ 1, [ 2, 6, 10 ] ], [ 1, [ 1, 8, 12 ] ], [ 1, [ 1, 7, 12 ] ], [ 1, [ 1, 7, 11 ] ], [ 1, [ 1, 6, 12 ] ], [ 1, [ 1, 6, 11 ] ], [ 1, [ 1, 6, 10 ] ], [ 1, [ 1, 5, 12 ] ], [ 1, [ 1, 5, 11 ] ], [ 1, [ 1, 5, 10 ] ], [ 1, [ 1, 5, 9 ] ] ] ], [ 1, [ [ 1, [ 13, 14, 15 ] ], [ 1, [ 9, 14, 15 ] ], [ 1, [ 9, 10, 15 ] ], [ 1, [ 9, 10, 11 ] ], [ 1, [ 5, 14, 15 ] ], [ 1, [ 5, 10, 15 ] ], [ 1, [ 5, 10, 11 ] ], [ 1, [ 5, 6, 15 ] ], [ 1, [ 5, 6, 11 ] ], [ 1, [ 5, 6, 7 ] ], [ 1, [ 1, 14, 15 ] ], [ 1, [ 1, 10, 15 ] ], [ 1, [ 1, 10, 11 ] ], [ 1, [ 1, 6, 15 ] ], [ 1, [ 1, 6, 11 ] ], [ 1, [ 1, 6, 7 ] ], [ 1, [ 1, 2, 15 ] ], [ 1, [ 1, 2, 11 ] ], [ 1, [ 1, 2, 7 ] ], [ 1, [ 1, 2, 3 ] ] ] ] ] gap> PrintArray(SCIntersectionForm(s2s2)); [ [ 0, 1 ], [ 1, 0 ] ] gap> c:=s2s2+s2s2; [SimplicialComplex Properties known: Dim, Facets, Name, VertexLabels, Vertices. Name="S^2_4xS^2_4#+-S^2_4xS^2_4" Dim=4 /SimplicialComplex] gap> PrintArray(SCIntersectionForm(c)); [ [ 0, -1, 0, 0 ], [ -1, 0, 0, 0 ], [ 0, 0, 0, -1 ], [ 0, 0, -1, 0 ] ]
For a more detailed description of functions related to bistellar flips as well as a very short introduction into the topic, see Chapter 9.
gap> beta4:=SCBdCrossPolytope(4);; gap> s3:=SCBdSimplex(4);; gap> SCEquivalent(beta4,s3); #I round 0, move: [ [ 2, 6, 7 ], [ 3, 4 ] ] [ 8, 25, 34, 17 ] #I round 1, move: [ [ 2, 7 ], [ 3, 4, 5 ] ] [ 8, 24, 32, 16 ] #I round 2, move: [ [ 2, 5 ], [ 3, 4, 8 ] ] [ 8, 23, 30, 15 ] #I round 3, move: [ [ 2 ], [ 3, 4, 6, 8 ] ] [ 7, 19, 24, 12 ] #I round 4, move: [ [ 6, 8 ], [ 1, 3, 4 ] ] [ 7, 18, 22, 11 ] #I round 5, move: [ [ 8 ], [ 1, 3, 4, 5 ] ] [ 6, 14, 16, 8 ] #I round 6, move: [ [ 5 ], [ 1, 3, 4, 7 ] ] [ 5, 10, 10, 5 ] #I SCReduceComplexEx: complexes are bistellarly equivalent. true gap> SCBistellarOptions.WriteLevel; 0 gap> SCBistellarOptions.WriteLevel:=1; 1 gap> SCEquivalent(beta4,s3); #I SCLibInit: made directory "~/PATH" for user library. #I 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 ]
For a more detailed description of functions related to simplicial blowups see Chapter 10.
gap> list:=SCLib.SearchByName("Kummer"); [ [ 7493, "4-dimensional Kummer variety (VT)" ] ] 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, [ ] ] ]
For a more detailed description of functions related to discrete normal surfaces and slicings see the Sections 2.4 and 2.5.
gap> # the boundary of the cyclic 4-polytope with 6 vertices gap> c:=SCBdCyclicPolytope(4,6); [SimplicialComplex Properties known: Dim, EulerCharacteristic, FacetsEx, HasBoundary, Homology,\ IsConnected, IsStronglyConnected, Name, NumFaces[], TopologicalType, Vertices. Name="Bd(C_4(6))" Dim=3 EulerCharacteristic=0 HasBoundary=false Homology=[ [ 0, [ ] ], [ 0, [ ] ], [ 0, [ ] ], [ 1, [ ] ] ] IsConnected=true IsStronglyConnected=true TopologicalType="S^3" /SimplicialComplex] gap> # slicing in between the odd and the even vertex labels, a polyhedral torus gap> sl:=SCSlicing(c,[[2,4,6],[1,3,5]]); [NormalSurface Properties known: ConnectedComponents, Dim, EulerCharacteristic, FVector,\ FacetsEx, Genus, IsConnected, IsOrientable, NSTriangulation, Name,\ TopologicalType, Vertices. Name="slicing [ [ 2, 4, 6 ], [ 1, 3, 5 ] ] of Bd(C_4(6))" Dim=2 FVector=[ 9, 18, 0, 9 ] EulerCharacteristic=0 IsOrientable=true TopologicalType="T^2" /NormalSurface] gap> sl.Homology; [ [ 0, [ ] ], [ 2, [ ] ], [ 1, [ ] ] ] gap> sl.Genus; 1 gap> sl.F; # the slicing constists of 9 quadrilaterals and 0 triangles [ 9, 18, 0, 9 ] gap> PrintArray(sl.Facets); [ [ [ 2, 1 ], [ 2, 3 ], [ 4, 1 ], [ 4, 3 ] ], [ [ 2, 1 ], [ 2, 3 ], [ 6, 1 ], [ 6, 3 ] ], [ [ 2, 1 ], [ 2, 5 ], [ 4, 1 ], [ 4, 5 ] ], [ [ 2, 1 ], [ 2, 5 ], [ 6, 1 ], [ 6, 5 ] ], [ [ 2, 3 ], [ 2, 5 ], [ 4, 3 ], [ 4, 5 ] ], [ [ 2, 3 ], [ 2, 5 ], [ 6, 3 ], [ 6, 5 ] ], [ [ 4, 1 ], [ 4, 3 ], [ 6, 1 ], [ 6, 3 ] ], [ [ 4, 1 ], [ 4, 5 ], [ 6, 1 ], [ 6, 5 ] ], [ [ 4, 3 ], [ 4, 5 ], [ 6, 3 ], [ 6, 5 ] ] ]
Further example computations can be found in the slides of various talks about simpcomp, available from the simpcomp homepage (https://github.com/simpcomp-team/simpcomp), and in Appendix A of [Spr11a].
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