simpcomp contains a library of simplicial complexes on few vertices, most of them (combinatorial) triangulations of manifolds and pseudomanifolds. The user can load these known triangulations from the library in order to study their properties or to construct new triangulations out of the known ones. For example, a user could determine the topological type of a given triangulation -- which can be quite tedious if done by hand -- by establishing a PL equivalence to a complex in the library.
Among other known triangulations, the library contains all of the vertex transitive triangulations of combinatorial manifolds with up to \(15\) vertices (for \(d \in \{ 2,3,9,10,11,12\}\)) and up to \(13\) vertices (for \(d \in \{ 4,5,6,7,8 \}\)) and all of the vertex transitive combinatorial pseudomanifolds with up to \(15\) vertices (for \(d=3\)) and up to \(13\) vertices (for \(d \in \{ 4,5,6,7 \} \)) classified by Frank Lutz that can be found on his ``Manifold Page'' http://www.math.tu-berlin.de/diskregeom/stellar/, along with some triangulations of sphere bundles and some bounded triangulated PL-manifolds.
See SCLib (13.1-2) for a naming convention used for the global library of simpcomp. Note: Another way of storing and loading complexes is provided by the functions SCExportIsoSig (6.2-2), SCExportToString (6.2-1) and SCFromIsoSig (6.2-3), see Section 6.2 for details.
‣ SCIsLibRepository( object ) | ( filter ) |
Returns: true or false upon success, fail otherwise.
Filter for the category of a library repository SCIsLibRepository used by the simpcomp library. The category SCLibRepository is derived from the category SCPropertyObject.
gap> SCIsLibRepository(SCLib); #the global library is stored in SCLib true
‣ SCLib | ( global variable ) |
The global variable SCLib contains the library object of the global library of simpcomp through which the user can access the library. The path to the global library is GAPROOT/pkg/simpcomp/complexes.
The naming convention in the global library is the following: complexes are usually named by their topological type. As usual, `S^d' denotes a \(d\)-sphere, `T' a torus, `x' the cartesian product, `~' the twisted product and `#' the connected sum. The Klein Bottle is denoted by `K' or `K^2'.
gap> SCLib;
[Simplicial complex library. Properties:
CalculateIndexAttributes=true
Number of complexes in library=648
IndexAttributes=[ "Name", "Dim", "F", "G", "H", "Chi", "Homology", "IsPM",
"IsManifold" ]
Loaded=true
Path="/home/jonathan/bin/gap-4.11.1/pkg/simpcomp/complexes/"
]
gap> SCLib.Size;
648
gap> SCLib.SearchByName("S^4~");
[ [ 203, "S^4~S^1 (VT)" ], [ 330, "S^4~S^1 (VT)" ], [ 332, "S^4~S^1 (VT)" ],
[ 395, "S^4~S^1 (VT)" ], [ 451, "S^4~S^1 (VT)" ], [ 452, "S^4~S^1 (VT)" ],
[ 453, "S^4~S^1 (VT)" ], [ 454, "S^4~S^1 (VT)" ], [ 455, "S^4~S^1 (VT)" ],
[ 458, "S^4~S^1 (VT)" ], [ 459, "S^4~S^1 (VT)" ], [ 460, "S^4~S^1 (VT)" ] ]
gap> SCLib.Load(last[1][1]);
<SimplicialComplex: S^4~S^1 (VT) | dim = 5 | n = 13>
‣ SCLibAdd( repository, complex[, name] ) | ( function ) |
Returns: true upon success, fail otherwise.
Adds a given simplicial complex complex to a given repository repository of type SCIsLibRepository. complex is saved to a file with suffix .sc in the repositories base path, where the file name is either formed from the optional argument name and the current time or taken from the name of the complex, if it is named.
gap> info:=InfoLevel(InfoSimpcomp);;
gap> SCInfoLevel(0);;
gap> myRepository:=SCLibInit("/tmp/repository");
[Simplicial complex library. Properties:
CalculateIndexAttributes=true
Number of complexes in library=0
IndexAttributes=[ "Name", "Dim", "F", "G", "H", "Chi", "Homology", "IsPM",
"IsManifold" ]
Loaded=true
Path="/tmp/repository/"
]
gap> complex1:=SCBdCrossPolytope(4);;
gap> SCLibAdd(myRepository,complex1);
true
gap> complex2:=SCBdCrossPolytope(4);;
gap> myRepository.Add(complex2);; # alternative syntax
gap> SCInfoLevel(info);;
‣ SCLibAllComplexes( repository ) | ( function ) |
Returns: list of entries of the form [ integer, string ] upon success, fail otherwise.
Returns a list with entries of the form [ ID, NAME ] of all the complexes in the given repository repository of type SCIsLibRepository.
gap> all:=SCLibAllComplexes(SCLib);; gap> all[1]; [ 1, "Moebius Strip" ] gap> Length(all); 648
‣ SCLibDelete( repository, id ) | ( function ) |
Returns: true upon success, fail otherwise.
Deletes the simplicial complex with the given id id from the given repository repository. Apart from deleting the complexes' index entry, the associated .sc file is also deleted.
gap> myRepository:=SCLibInit("/tmp/repository");
[Simplicial complex library. Properties:
CalculateIndexAttributes=true
Number of complexes in library=2
IndexAttributes=[ "Name", "Dim", "F", "G", "H", "Chi", "Homology", "IsPM",
"IsManifold" ]
Loaded=true
Path="/tmp/repository/"
]
gap> SCLibAdd(myRepository,SCSimplex(2));;
gap> SCLibDelete(myRepository,1);
true
‣ SCLibDetermineTopologicalType( [repository, ]complex ) | ( function ) |
Returns: simplicial complex of type SCSimplicialComplex or a list of integers upon success, fail otherwise.
Tries to determine the topological type of a given complex complex by first looking for complexes with matching homology in the library repository repository (if no repository is passed, the global repository SCLib is used) and either returns a simplicial complex object (that is combinatorially isomorphic to the complex given) or a list of library ids of complexes in the library with the same homology as the complex provided.
The ids obtained in this way can then be used to compare the corresponding complexes with complex via the function SCEquivalent (9.2-2).
If complex is a combinatorial manifold of dimension \(1\) or \(2\) its topological type is computed, stored to the property TopologicalType and complex is returned.
If no complexes with matching homology can be found, the empty set is returned.
gap> c:=SCFromFacets([[1,2,3],[1,2,6],[1,3,5],[1,4,5],[1,4,6],
[2,3,4],[2,4,5],[2,5,6],[3,4,6],[3,5,6]]);;
gap> SCLibDetermineTopologicalType(c);
<SimplicialComplex: unnamed complex 250 | dim = 2 | n = 6>
‣ SCLibFlush( repository, confirm ) | ( function ) |
Returns: true upon success, fail otherwise.
Completely empties a given repository repository. The index and all simplicial complexes in this repository are deleted. The second argument, confirm, must be the string "yes" in order to confirm the deletion.
gap> myRepository:=SCLibInit("/tmp/repository");;
gap> SCLibFlush(myRepository,"yes");
#I SCLibInit: invalid parameters.
true
‣ SCLibInit( dir ) | ( function ) |
Returns: library repository of type SCLibRepository upon success, fail otherwise.
This function initializes a library repository object for the given directory dir (which has to be provided in form of a GAP object of type String or Directory) and returns that library repository object in case of success. The returned object then provides a mean to access the library repository via the SCLib-functions of simpcomp.
The global library repository of simpcomp is loaded automatically at startup and is stored in the variable SCLib. User repositories can be created by calling SCLibInit with a desired destination directory. Note that each repository must reside in a different path since otherwise data may get lost.
The function first tries to load the repository index for the given directory to rebuild it (by calling SCLibUpdate) if loading the index fails. The library index of a library repository is stored in its base path in the XML file complexes.idx, the complexes are stored in files with suffix .sc, also in XML format.
gap> myRepository:=SCLibInit("/tmp/repository");
#I SCLibInit: made directory "/tmp/repository/" for user library.
#I SCIntFunc.SCLibInit: index not found -- trying to reconstruct it.
#I SCLibUpdate: rebuilding index for /tmp/repository/.
#I SCLibUpdate: rebuilding index done.
[Simplicial complex library. Properties:
CalculateIndexAttributes=true
Number of complexes in library=0
IndexAttributes=[ "Name", "Dim", "F", "G", "H", "Chi", "Homology", "IsPM",
"IsManifold" ]
Loaded=true
Path="/tmp/repository/"
]
‣ SCLibIsLoaded( repository ) | ( function ) |
Returns: true or false upon succes, fail otherwise.
Returns true when a given library repository repository is in loaded state. This means that the directory of this repository is accessible and a repository index file for this repository exists in the repositories' path. If this is not the case false is returned.
gap> SCLibIsLoaded(SCLib); true gap> SCLib.IsLoaded; true
‣ SCLibSearchByAttribute( repository, expr ) | ( function ) |
Returns: A list of items of the form [ integer, string ] upon success, fail otherwise.
Searches a given repository repository for complexes for which the boolean expression expr, passed as string, evaluates to true and returns a list of complexes with entries of the form [ID, NAME] or fail upon error. The expression may use all GAP functions and can access all the indexed attributes of the complexes in the given repository for the query. The standard attributes are: Dim (Dimension), F (f-vector), G (g-vector), H (h-vector), Chi (Euler characteristic), Homology, Name, IsPM, IsManifold. See SCLib for the set of indexed attributes of the global library of simpcomp.
gap> SCLibSearchByAttribute(SCLib,"Dim=4 and F[3]=Binomial(F[1],3)");
[ [ 16, "CP^2 (VT)" ], [ 520, "K3_16" ] ]
gap> SCLib.SearchByAttribute("Dim=4 and F[3]=Binomial(F[1],3)");
[ [ 16, "CP^2 (VT)" ], [ 520, "K3_16" ] ]
‣ SCLibSearchByName( repository, name ) | ( function ) |
Returns: A list of items of the form [ integer, string ] upon success, fail otherwise.
Searches a given repository repository for complexes that contain the string name as a substring of their name attribute and returns a list of the complexes found with entries of the form [ID, NAME]. See SCLib (13.1-2) for a naming convention used for the global library of simpcomp.
gap> SCLibSearchByName(SCLib,"K3");
[ [ 520, "K3_16" ], [ 539, "K3_17" ] ]
gap> SCLib.SearchByName("K3"); #alternative syntax
[ [ 520, "K3_16" ], [ 539, "K3_17" ] ]
gap> SCLib.SearchByName("S^4x"); #search for products with S^4
[ [ 282, "S^4xS^1 (VT)" ], [ 329, "S^4xS^1 (VT)" ], [ 331, "S^4xS^1 (VT)" ] ]
‣ SCLibSize( repository ) | ( function ) |
Returns: integer upon success, fail otherwise.
Returns the number of complexes contained in the given repository repository. Fails if the library repository was not previously loaded with SCLibInit.
gap> SCLibSize(SCLib); #SCLib is the repository of the global library 648
‣ SCLibUpdate( repository[, recalc] ) | ( function ) |
Returns: library repository of type SCLibRepository upon success, fail otherwise.
Recreates the index of a given repository (either via a repository object or a base path of a repository repository) by scanning the base path for all .sc files containing simplicial complexes of the repository. Returns a repository object with the newly created index on success or fail in case of an error. The optional boolean argument recalc forces simpcomp to recompute all the indexed properties (such as f-vector, homology, etc.) of the simplicial complexes in the repository if set to true.
gap> myRepository:=SCLibInit("/tmp/repository");;
gap> SCLibUpdate(myRepository);
#I SCLibUpdate: rebuilding index for /tmp/repository/.
#I SCLibUpdate: rebuilding index done.
[Simplicial complex library. Properties:
CalculateIndexAttributes=true
Number of complexes in library=0
IndexAttributes=[ "Name", "Dim", "F", "G", "H", "Chi", "Homology", "IsPM",
"IsManifold" ]
Loaded=true
Path="/tmp/repository/"
]
‣ SCLibStatus( repository ) | ( function ) |
Returns: library repository of type SCLibRepository upon success, fail otherwise.
Lets GAP print the status of a given library repository repository. IndexAttributes is the list of attributes indexed for this repository. If CalculateIndexAttributes is true, the index attributes for a complex added to the library are calculated automatically upon addition of the complex, otherwise this is left to the user and only pre-calculated attributes are indexed.
gap> SCLibStatus(SCLib); [Simplicial complex library. Properties: CalculateIndexAttributes=true Number of complexes in library=648 IndexAttributes=[ "Name", "Dim", "F", "G", "H", "Chi", "Homology", "IsPM", "IsManifold" ] Loaded=true Path="/home/jonathan/bin/gap-4.11.1/pkg/simpcomp/complexes/" ]
This section contains a description of the input/output-functionality provided by simpcomp. The package provides the functionality to save and load simplicial complexes (and their known properties) to, respectively from files in XML format. Furthermore, it provides the user with functions to export simplicial complexes into polymake format (for this format there also exists rudimentary import functionality), as JavaView geometry or in form of a LaTeX table. For importing more complex polymake data the package polymaking [R\t13] can be used.
‣ SCLoad( filename ) | ( function ) |
Returns: simplicial complex of type SCSimplicialComplex upon success, fail otherwise.
Loads a simplicial complex stored in a binary format (using IO_Pickle) from a file specified in filename (as string). If filename does not end in .scb, this suffix is appended to the file name.
gap> c:=SCBdSimplex(3);;
gap> SCSave(c,"/tmp/bddelta3");
true
gap> d:=SCLoad("/tmp/bddelta3");
<SimplicialComplex: S^2_4 | dim = 2 | n = 4>
gap> c=d;
true
‣ SCLoadXML( filename ) | ( function ) |
Returns: simplicial complex of type SCSimplicialComplex upon success, fail otherwise.
Loads a simplicial complex stored in XML format from a file specified in filename (as string). If filename does not end in .sc, this suffix is appended to the file name.
gap> c:=SCBdSimplex(3);;
gap> SCSaveXML(c,"/tmp/bddelta3");
true
gap> d:=SCLoadXML("/tmp/bddelta3");
<SimplicialComplex: S^2_4 | dim = 2 | n = 4>
gap> c=d;
true
‣ SCSave( complex, filename ) | ( function ) |
Returns: true upon success, fail otherwise.
Saves a simplicial complex in a binary format (using IO_Pickle) to a file specified in filename (as string). If filename does not end in .scb, this suffix is appended to the file name.
gap> c:=SCBdSimplex(3);; gap> SCSave(c,"/tmp/bddelta3"); true
‣ SCSaveXML( complex, filename ) | ( function ) |
Returns: true upon success, fail otherwise.
Saves a simplicial complex complex to a file specified by filename (as string) in XML format. If filename does not end in .sc, this suffix is appended to the file name.
gap> c:=SCBdSimplex(3);; gap> SCSaveXML(c,"/tmp/bddelta3"); true
‣ SCExportMacaulay2( complex, ring, filename[, alphalabels] ) | ( function ) |
Returns: true upon success, fail otherwise.
Exports the facet list of a given simplicial complex complex in Macaulay2 format to a file specified by filename. The argument ring can either be the ring of integers (specified by Integers) or the ring of rationals (sepcified by Rationals). The optional boolean argument alphalabels labels the complex with characters from \(a, \dots ,z\) in the exported file if a value of true is supplied, while the standard labeling of the vertices is \(v_1, \dots ,v_n\) where \(n\) is the number of vertices of complex. If complex has more than \(26\) vertices, the argument alphalabels is ignored.
gap> c:=SCBdCrossPolytope(4);; gap> SCExportMacaulay2(c,Integers,"/tmp/bdbeta4.m2"); true
‣ SCExportPolymake( complex, filename ) | ( function ) |
Returns: true upon success, fail otherwise.
Exports the facet list with vertex labels of a given simplicial complex complex in polymake format to a file specified by filename. Currently, only the export in the format of polymake version 2.3 is supported.
gap> c:=SCBdCrossPolytope(4);; gap> SCExportPolymake(c,"/tmp/bdbeta4.poly"); true
‣ SCImportPolymake( filename ) | ( function ) |
Returns: simplicial complex of type SCSimplicialComplex upon success, fail otherwise.
Imports the facet list of a topaz polymake file specified by filename (discarding any vertex labels) and creates a simplicial complex object from these facets.
gap> c:=SCBdCrossPolytope(4);;
gap> SCExportPolymake(c,"/tmp/bdbeta4.poly");
true
gap> d:=SCImportPolymake("/tmp/bdbeta4.poly");
<SimplicialComplex: polymake import '/tmp/bdbeta4.poly' | dim = 3 | n = 8>
gap> c=d;
true
‣ SCExportLatexTable( complex, filename, itemsperline ) | ( function ) |
Returns: true on success, fail otherwise.
Exports the facet list of a given simplicial complex complex (or any list given as first argument) in form of a LaTeX table to a file specified by filename. The argument itemsperline specifies how many columns the exported table should have. The faces are exported in the format \(\langle v_1,\dots,v_k \rangle\).
gap> c:=SCBdSimplex(5);; gap> SCExportLatexTable(c,"/tmp/bd5simplex.tex",5); true
‣ SCExportJavaView( complex, file, coords ) | ( function ) |
Returns: true on success, fail otherwise.
Exports the 2-skeleton of the given simplicial complex complex (or the facets if the complex is of dimension 2 or less) in JavaView format (file name suffix .jvx) to a file specified by filename (as string). The list coords must contain a \(3\)-tuple of real coordinates for each vertex of complex, either as tuple of length three containing the coordinates (Warning: as GAP only has rudimentary support for floating point values, currently only integer numbers can be used as coordinates when providing coords as list of \(3\)-tuples) or as string of the form "x.x y.y z.z" with decimal numbers x.x, y.y, z.z for the three coordinates (i.e. "1.0 0.0 0.0").
gap> coords:=[[1,0,0],[0,1,0],[0,0,1]];; gap> SCExportJavaView(SCBdSimplex(2),"/tmp/triangle.jvx",coords); true
‣ SCExportRecognizer( complex, filename ) | ( function ) |
Returns: true upon success, fail otherwise.
Exports the gluings of the tetrahedra of a given combinatorial \(3\)-manifold complex in a format compatible with Matveev's \(3\)-manifold software Recognizer.
gap> c:=SCBdCrossPolytope(4);; gap> SCExportRecognizer(c,"/tmp/bdbeta4.mv"); true
‣ SCExportSnapPy( complex, filename ) | ( function ) |
Returns: true upon success, fail otherwise.
Exports the facet list and orientability of a given combinatorial \(3\)-pseudomanifold complex in SnapPy format to a file specified by filename.
gap> SCLib.SearchByAttribute("Dim=3 and F=[8,28,56,28]");
[ [ 8, "PM^3 - TransitiveGroup(8,43), No. 1" ] ]
gap> c:=SCLib.Load(last[1][1]);;
gap> SCExportSnapPy(c,"/tmp/M38.tri");
true
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