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<a id="X78CC1D0D809E5D2E" name="X78CC1D0D809E5D2E"></a>
<div class="ChapSects"><a href="chap17.html#X78CC1D0D809E5D2E">17 A demo session with simpcomp</a>
<div class="ContSect"> <a href="chap17.html#X802AC56478332D59">17.1 Creating a <code class="code">SCSimplicialComplex</code> object</a>

</div>
<div class="ContSect"> <a href="chap17.html#X7F2BA767781B3216">17.2 Working with a <code class="code">SCSimplicialComplex</code> object</a>

</div>
<div class="ContSect"> <a href="chap17.html#X81109FCF7A2BAD69">17.3 Calculating properties of a <code class="code">SCSimplicialComplex</code> object</a>

</div>
<div class="ContSect"> <a href="chap17.html#X833CFE367E7591A0">17.4 Creating new complexes from a <code class="code">SCSimplicialComplex</code> object</a>

</div>
<div class="ContSect"> <a href="chap17.html#X7F91D2307D7CE8C8">17.5 Homology related calculations</a>

</div>
<div class="ContSect"> <a href="chap17.html#X82F1CE7A79A3CA47">17.6 Bistellar flips</a>

</div>
<div class="ContSect"> <a href="chap17.html#X7E1BE673859191F0">17.7 Simplicial blowups</a>

</div>
<div class="ContSect"> <a href="chap17.html#X7B5C29557E8E92EA">17.8 Discrete normal surfaces and slicings</a>

</div>
</div>

<h3>17 A demo session with simpcomp</h3>

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 <a href="chap4.html#X840691C285AB3AAD">4</a> through <a href="chap15.html#X8308D685809A4E2F">15</a> for further functions provided by simpcomp.

<a id="X802AC56478332D59" name="X802AC56478332D59"></a>

<h4>17.1 Creating a <code class="code">SCSimplicialComplex</code> object</h4>

Simplicial complex objects can either be created from a facet list (complex <code class="code">c1</code> below), orbit representatives together with a permutation group (complex <code class="code">c2</code>) or difference cycles (complex <code class="code">c3</code>, see Section <a href="chap6.html#X7A93E4B08536E2C8">6.1</a>), from a function generating triangulations of standard complexes (complex <code class="code">c4</code>, see Section <a href="chap6.html#X79072405786FEA0B">6.3</a>) or from a function constructing infinite series for combinatorial (pseudo)manifolds (complexes <code class="code">c5</code>, <code class="code">c6</code>, <code class="code">c7</code>, see Section <a href="chap6.html#X814FE0267D7C54A9">6.4</a> and the function prefix <code class="code">SCSeries...</code>). There are also functions creating new simplicial complexes from old, see Section <a href="chap6.html#X7F4308DB7C3699D1">6.6</a>, which will be described in the next sections.

<div class="example"><pre>
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
</pre></div>

<a id="X7F2BA767781B3216" name="X7F2BA767781B3216"></a>

<h4>17.2 Working with a <code class="code">SCSimplicialComplex</code> object</h4>

As described in Section <a href="chap3.html#X82FAB51B7C625240">3.1</a> there are two several ways of accessing an object of type <code class="code">SCSimplicialComplex</code>. 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

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

is equivalent to

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

<a id="X81109FCF7A2BAD69" name="X81109FCF7A2BAD69"></a>

<h4>17.3 Calculating properties of a <code class="code">SCSimplicialComplex</code> object</h4>

simpcomp provides a variety of functions for calculating properties of simplicial complexes, see Section <a href="chap6.html#X81CE90127800B91A">6.9</a>. All these properties are only calculated once and stored in the <code class="code">SCSimplicialComplex</code> object.

<div class="example"><pre>
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
</pre></div>

<a id="X833CFE367E7591A0" name="X833CFE367E7591A0"></a>

<h4>17.4 Creating new complexes from a <code class="code">SCSimplicialComplex</code> object</h4>

As already mentioned, there is the possibility to generate new objects of type <code class="code">SCSimplicialComplex</code> from existing ones using standard constructions. The functions used in this section are described in more detail in Section <a href="chap6.html#X7F4308DB7C3699D1">6.6</a>.

<div class="example"><pre>
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]
</pre></div>

<a id="X7F91D2307D7CE8C8" name="X7F91D2307D7CE8C8"></a>

<h4>17.5 Homology related calculations</h4>

simpcomp relies on the GAP package homology <a href="chapBib.html#biBDumas04Homology">[DHSW11]</a> for its homology computations but provides further (co-)homology related functions, see Chapter <a href="chap8.html#X7B0C706A848A2542">8</a>.

<div class="example"><pre>
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 ] ]
</pre></div>

<a id="X82F1CE7A79A3CA47" name="X82F1CE7A79A3CA47"></a>

<h4>17.6 Bistellar flips</h4>

For a more detailed description of functions related to bistellar flips as well as a very short introduction into the topic, see Chapter <a href="chap9.html#X82F1CE7A79A3CA47">9</a>.

<div class="example"><pre>
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 ]
 </pre></div>

<a id="X7E1BE673859191F0" name="X7E1BE673859191F0"></a>

<h4>17.7 Simplicial blowups</h4>

For a more detailed description of functions related to simplicial blowups see Chapter <a href="chap10.html#X7E1BE673859191F0">10</a>.

<div class="example"><pre>
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, [ ] ] ]
 </pre></div>

<a id="X7B5C29557E8E92EA" name="X7B5C29557E8E92EA"></a>

<h4>17.8 Discrete normal surfaces and slicings</h4>

For a more detailed description of functions related to discrete normal surfaces and slicings see the Sections <a href="chap2.html#X7BE7221B7C38B27D">2.4</a> and <a href="chap2.html#X86275D5979B4B531">2.5</a>.

<div class="example"><pre>

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 ] ] ]
</pre></div>

Further example computations can be found in the slides of various talks about simpcomp, available from the simpcomp homepage (<code class="code">https://github.com/simpcomp-team/simpcomp</code>), and in Appendix A of <a href="chapBib.html#biBSpreer10Diss">[Spr11a]</a>.

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