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### 4 Functions and operations for the GAP object type SCPolyhedralComplex

In the following all operations for the GAP object type SCPolyhedralComplex are listed. I. e. for the following operations only one method is implemented to deal with all geometric objects derived from this object type.

#### 4.1 Computing properties of objects of type SCPolyhedralComplex

The following functions compute basic properties of objects of type SCPolyhedralComplex (and thus also of objects of type SCSimplicialComplex and SCNormalSurface). None of these functions alter the complex. All properties are returned as immutable objects (this ensures data consistency of the cached properties of a simplicial complex). Use ShallowCopy or the internal simpcomp function SCIntFunc.DeepCopy to get a mutable copy.

Note: every object is internally stored with the standard vertex labeling from 1 to n and a maptable to restore the original vertex labeling. Thus, we have to relabel some of the complex properties (facets, etc...) whenever we want to return them to the user. As a consequence, some of the functions exist twice, one of them with the appendix "Ex". These functions return the standard labeling whereas the other ones relabel the result to the original labeling.

##### 4.1-1 SCFacets
 ‣ SCFacets( complex ) ( method )

Returns: a facet list upon success, fail otherwise.

Returns the facets of a simplicial complex in the original vertex labeling.

gap> c:=SC([[2,3],[3,4],[4,2]]);;
gap> SCFacets(c);
[ [ 2, 3 ], [ 2, 4 ], [ 3, 4 ] ]

##### 4.1-2 SCFacetsEx
 ‣ SCFacetsEx( complex ) ( method )

Returns: a facet list upon success, fail otherwise.

Returns the facets of a simplicial complex as they are stored, i. e. with standard vertex labeling from 1 to n.

gap> c:=SC([[2,3],[3,4],[4,2]]);;
gap> SCFacetsEx(c);
[ [ 1, 2 ], [ 1, 3 ], [ 2, 3 ] ]

##### 4.1-3 SCVertices
 ‣ SCVertices( complex ) ( method )

Returns: a list of vertex labels of complex upon success, fail otherwise.

Returns the vertex labels of a simplicial complex complex.

gap> sphere:=SC([["x",45,[1,1]],["x",45,["b",3]],["x",[1,1],
["b",3]],[45,[1,1],["b",3]]]);;
gap> SCVerticesEx(sphere);
[ 1 .. 4 ]
gap> SCVertices(sphere);
[ 45, [ 1, 1 ], "x", [ "b", 3 ] ]

##### 4.1-4 SCVerticesEx
 ‣ SCVerticesEx( complex ) ( method )

Returns: [ 1, ... , n ] upon success, fail otherwise.

Returns [1, ... , n ], where n is the number of vertices of a simplicial complex complex.

gap> c:=SC([[1,4,5],[4,9,8],[12,13,14,15,16,17]]);;
gap> SCVerticesEx(c);
[ 1 .. 11 ]

#### 4.2 Vertex labelings and label operations

This section focuses on functions operating on the labels of a complex such as the name or the vertex labeling.

Internally, simpcomp uses the standard labeling [1, ... , n]. It is recommended to use simple vertex labels like integers and, whenever possible, the standard labeling, see also SCRelabelStandard (4.2-7).

##### 4.2-1 SCLabelMax
 ‣ SCLabelMax( complex ) ( method )

Returns: vertex label of complex (an integer, a short list, a character, a short string) upon success, fail otherwise.

The maximum over all vertex labels is determined by the GAP function MaximumList.

gap> c:=SCBdSimplex(3);;
gap> SCRelabel(c,[10,100,100000,3500]);;
gap> SCLabelMax(c);
100000

gap> c:=SCBdSimplex(3);;
gap> SCRelabel(c,["a","bbb",5,[1,1]]);;
gap> SCLabelMax(c);
"bbb"

##### 4.2-2 SCLabelMin
 ‣ SCLabelMin( complex ) ( method )

Returns: vertex label of complex (an integer, a short list, a character, a short string) upon success, fail otherwise.

The minimum over all vertex labels is determined by the GAP function MinimumList.

gap> c:=SCBdSimplex(3);;
gap> SCRelabel(c,[10,100,100000,3500]);;
gap> SCLabelMin(c);
10

gap> c:=SCBdSimplex(3);;
gap> SCRelabel(c,["a","bbb",5,[1,1]]);;
gap> SCLabelMin(c);
5

##### 4.2-3 SCLabels
 ‣ SCLabels( complex ) ( method )

Returns: a list of vertex labels of complex (a list of integers, short lists, characters, short strings, ...) upon success, fail otherwise.

Returns the vertex labels of complex as a list. This is a synonym of SCVertices (4.1-3).

gap> c:=SCFromFacets(Combinations(["a","b","c","d"],3));;
gap> SCLabels(c);
[ "a", "b", "c", "d" ]

##### 4.2-4 SCName
 ‣ SCName( complex ) ( operation )

Returns: a string upon success, fail otherwise.

Returns the name of a simplicial complex complex.

gap> c:=SCBdSimplex(5);;
gap> SCName(c);
"S^4_6"

gap> c:=SC([[1,2],[2,3],[3,1]]);;
gap> SCName(c);
"unnamed complex 2"

##### 4.2-5 SCReference
 ‣ SCReference( complex ) ( operation )

Returns: a string upon success, fail otherwise.

Returns a literature reference of a polyhedral complex complex.

gap> SCReference(c);
"F.H.Lutz: 'The Manifold Page', http://www.math.tu-berlin.de/diskregeom/stella\
r/"
gap> c:=SC([[1,2],[2,3],[3,1]]);;
gap> SCReference(c);
#I  SCReference: complex lacks reference.
fail

##### 4.2-6 SCRelabel
 ‣ SCRelabel( complex, maptable ) ( method )

Returns: true upon success, fail otherwise.

maptable has to be a list of length n where n is the number of vertices of complex. The function maps the i-th entry of maptable to the i-th entry of the current vertex labels. If complex has the standard vertex labeling [1, ... , n] the vertex label i is mapped to maptable[i].

Note that the elements of maptable must admit a total ordering. Hence, following Section 4.11 of the GAP manual, they must be members of one of the following families: rationals IsRat, cyclotomics IsCyclotomic, finite field elements IsFFE, permutations IsPerm, booleans IsBool, characters IsChar and lists (strings) IsList.

Internally the property SCVertices'' of complex is replaced by maptable.

gap> list:=SCLib.SearchByAttribute("F[1]=12");;
gap> SCVertices(c);
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ]
gap> SCRelabel(c,["a","b","c","d","e","f","g","h","i","j","k","l"]);
true
gap> SCLabels(c);
[ "a", "b", "c", "d", "e", "f", "g", "h", "i", "j", "k", "l" ]

##### 4.2-7 SCRelabelStandard
 ‣ SCRelabelStandard( complex ) ( method )

Returns: true upon success, fail otherwise.

Maps vertex labels v_1 , ... , v_n of complex to [1 , ... , n]. Internally the property "SCVertices" is replaced by [1 , ... , n].

gap> list:=SCLib.SearchByAttribute("F[1]=12");;
gap> SCRelabel(c,[4..15]);
true
gap> SCVertices(c);
[ 4 .. 15 ]
gap> SCRelabelStandard(c);
true
gap> SCLabels(c);
[ 1 .. 12 ]

##### 4.2-8 SCRelabelTransposition
 ‣ SCRelabelTransposition( complex, pair ) ( method )

Returns: true upon success, fail otherwise.

Permutes vertex labels of a single pair of vertices. pair has to be a list of length 2 and a sublist of the property SCVertices''.

The function is equivalent to SCRelabel (4.2-6) with maptable = [ SCVertices[1] , ... , SCVertices[j] , ... , SCVertices[i] , dots , SCVertices[n]] if pair = [ SCVertices[j] , SCVertices[i]], j ≤ i, j ≠ i.

gap> c:=SCBdSimplex(3);;
gap> SCVertices(c);
[ 1 .. 4 ]
gap> SCRelabelTransposition(c,[1,2]);;
gap> SCLabels(c);
[ 2, 1, 3, 4 ]

##### 4.2-9 SCRename
 ‣ SCRename( complex, name ) ( method )

Returns: true upon success, fail otherwise.

Renames a polyhedral complex. The argument name has to be given in form of a string.

gap> c:=SCBdSimplex(5);;
gap> SCName(c);
"S^4_6"
gap> SCRename(c,"mySphere");
true
gap> SCName(c);
"mySphere"

##### 4.2-10 SCSetReference
 ‣ SCSetReference( complex, ref ) ( method )

Returns: true upon success, fail otherwise.

Sets the literature reference of a polyhedral complex. The argument ref has to be given in form of a string.

gap> c:=SCBdSimplex(5);;
gap> SCReference(c);
#I  SCReference: complex lacks reference.
fail
gap> SCSetReference(c,"my 5-sphere in my cool paper");
true
gap> SCReference(c);
"my 5-sphere in my cool paper"

##### 4.2-11 SCUnlabelFace
 ‣ SCUnlabelFace( complex, face ) ( method )

Returns: a list upon success, fail otherwise.

Computes the standard labeling of face in complex.

gap> c:=SCBdSimplex(3);;
gap> SCRelabel(c,["a","bbb",5,[1,1]]);;
gap> SCUnlabelFace(c,["a","bbb",5]);
[ 1, 2, 3 ]

#### 4.3 Operations on objects of type SCPolyhedralComplex

The following functions perform operations on objects of type SCPolyhedralComplex and all of its subtypes. Most of them return simplicial complexes. Thus, this section is closely related to the Sections 6.6 (for objects of type SCSimplicialComplex), ''Generate new complexes from old''. However, the data generated here is rather seen as an intrinsic attribute of the original complex and not as an independent complex.

##### 4.3-1 SCAntiStar
 ‣ SCAntiStar( complex, face ) ( method )

Returns: simplicial complex of type SCSimplicialComplex upon success, fail otherwise .

Computes the anti star of face (a face given as a list of vertices or a scalar interpreted as vertex) in complex, i. e. the complement of face in complex.

gap> SCLib.SearchByName("RP^2");
[ [ 3, "RP^2 (VT)" ], [ 645, "RP^2xS^1" ] ]
gap> SCVertices(rp2);
[ 1, 2, 3, 4, 5, 6 ]
gap> SCAntiStar(rp2,1);
[SimplicialComplex

Properties known: Dim, FacetsEx, Name, Vertices.

Name="ast([ 1 ]) in RP^2 (VT)"
Dim=2

/SimplicialComplex]
gap> last.Facets;
[ [ 2, 3, 4 ], [ 2, 4, 5 ], [ 2, 5, 6 ], [ 3, 4, 6 ], [ 3, 5, 6 ] ]

 ‣ SCLink( complex, face ) ( method )

Returns: simplicial complex of type SCSimplicialComplex upon success, fail otherwise.

Computes the link of face (a face given as a list of vertices or a scalar interpreted as vertex) in a polyhedral complex complex, i. e. all facets containing face, reduced by face. if complex is pure, the resulting complex is of dimension dim(complex) - dim(face) -1. If face is not a face of complex the empty complex is returned.

gap> SCLib.SearchByName("RP^2");
[ [ 3, "RP^2 (VT)" ], [ 645, "RP^2xS^1" ] ]
gap> SCVertices(rp2);
[ 1, 2, 3, 4, 5, 6 ]
[SimplicialComplex

Properties known: Dim, FacetsEx, Name, Vertices.

Name="lk([ 1 ]) in RP^2 (VT)"
Dim=1

/SimplicialComplex]
gap> last.Facets;
[ [ 2, 3 ], [ 2, 6 ], [ 3, 5 ], [ 4, 5 ], [ 4, 6 ] ]

 ‣ SCLinks( complex, k ) ( method )

Returns: a list of simplicial complexes of type SCSimplicialComplex upon success, fail otherwise.

Computes the link of all k-faces of the polyhedral complex complex and returns them as a list of simplicial complexes. Internally calls SCLink (4.3-2) for every k-face of complex.

gap> c:=SCBdSimplex(4);;
[ [SimplicialComplex

Properties known: Dim, FacetsEx, Name, Vertices.

Name="lk([ 1 ]) in S^3_5"
Dim=2

/SimplicialComplex], [SimplicialComplex

Properties known: Dim, FacetsEx, Name, Vertices.

Name="lk([ 2 ]) in S^3_5"
Dim=2

/SimplicialComplex], [SimplicialComplex

Properties known: Dim, FacetsEx, Name, Vertices.

Name="lk([ 3 ]) in S^3_5"
Dim=2

/SimplicialComplex], [SimplicialComplex

Properties known: Dim, FacetsEx, Name, Vertices.

Name="lk([ 4 ]) in S^3_5"
Dim=2

/SimplicialComplex], [SimplicialComplex

Properties known: Dim, FacetsEx, Name, Vertices.

Name="lk([ 5 ]) in S^3_5"
Dim=2

/SimplicialComplex] ]
[ [SimplicialComplex

Properties known: Dim, FacetsEx, Name, Vertices.

Name="lk([ 1, 2 ]) in S^3_5"
Dim=1

/SimplicialComplex], [SimplicialComplex

Properties known: Dim, FacetsEx, Name, Vertices.

Name="lk([ 1, 3 ]) in S^3_5"
Dim=1

/SimplicialComplex], [SimplicialComplex

Properties known: Dim, FacetsEx, Name, Vertices.

Name="lk([ 1, 4 ]) in S^3_5"
Dim=1

/SimplicialComplex], [SimplicialComplex

Properties known: Dim, FacetsEx, Name, Vertices.

Name="lk([ 1, 5 ]) in S^3_5"
Dim=1

/SimplicialComplex], [SimplicialComplex

Properties known: Dim, FacetsEx, Name, Vertices.

Name="lk([ 2, 3 ]) in S^3_5"
Dim=1

/SimplicialComplex], [SimplicialComplex

Properties known: Dim, FacetsEx, Name, Vertices.

Name="lk([ 2, 4 ]) in S^3_5"
Dim=1

/SimplicialComplex], [SimplicialComplex

Properties known: Dim, FacetsEx, Name, Vertices.

Name="lk([ 2, 5 ]) in S^3_5"
Dim=1

/SimplicialComplex], [SimplicialComplex

Properties known: Dim, FacetsEx, Name, Vertices.

Name="lk([ 3, 4 ]) in S^3_5"
Dim=1

/SimplicialComplex], [SimplicialComplex

Properties known: Dim, FacetsEx, Name, Vertices.

Name="lk([ 3, 5 ]) in S^3_5"
Dim=1

/SimplicialComplex], [SimplicialComplex

Properties known: Dim, FacetsEx, Name, Vertices.

Name="lk([ 4, 5 ]) in S^3_5"
Dim=1

/SimplicialComplex] ]

##### 4.3-4 SCStar
 ‣ SCStar( complex, face ) ( method )

Returns: simplicial complex of type SCSimplicialComplex upon success, fail otherwise .

Computes the star of face (a face given as a list of vertices or a scalar interpreted as vertex) in a polyhedral complex complex, i. e. the set of facets of complex that contain face.

gap> SCLib.SearchByName("RP^2");
[ [ 3, "RP^2 (VT)" ], [ 645, "RP^2xS^1" ] ]
gap> SCVertices(rp2);
[ 1, 2, 3, 4, 5, 6 ]
gap> SCStar(rp2,1);
[SimplicialComplex

Properties known: Dim, FacetsEx, Name, Vertices.

Name="star([ 1 ]) in RP^2 (VT)"
Dim=2

/SimplicialComplex]
gap> last.Facets;
[ [ 1, 2, 3 ], [ 1, 2, 6 ], [ 1, 3, 5 ], [ 1, 4, 5 ], [ 1, 4, 6 ] ]

##### 4.3-5 SCStars
 ‣ SCStars( complex, k ) ( method )

Returns: a list of simplicial complexes of type SCSimplicialComplex upon success, fail otherwise.

Computes the star of all k-faces of the polyhedral complex complex and returns them as a list of simplicial complexes. Internally calls SCStar (4.3-4) for every k-face of complex.

gap> SCLib.SearchByName("T^2"){[1..6]};
[ [ 4, "T^2 (VT)" ], [ 5, "T^2 (VT)" ], [ 9, "T^2 (VT)" ], [ 10, "T^2 (VT)" ],
[ 18, "T^2 (VT)" ], [ 20, "(T^2)#2" ] ]
gap> torus:=SCLib.Load(last[1][1]);; # the minimal 7-vertex torus
gap> SCStars(torus,0); # 7 2-discs as vertex stars
[ [SimplicialComplex

Properties known: Dim, FacetsEx, Name, Vertices.

Name="star([ 1 ]) in T^2 (VT)"
Dim=2

/SimplicialComplex], [SimplicialComplex

Properties known: Dim, FacetsEx, Name, Vertices.

Name="star([ 2 ]) in T^2 (VT)"
Dim=2

/SimplicialComplex], [SimplicialComplex

Properties known: Dim, FacetsEx, Name, Vertices.

Name="star([ 3 ]) in T^2 (VT)"
Dim=2

/SimplicialComplex], [SimplicialComplex

Properties known: Dim, FacetsEx, Name, Vertices.

Name="star([ 4 ]) in T^2 (VT)"
Dim=2

/SimplicialComplex], [SimplicialComplex

Properties known: Dim, FacetsEx, Name, Vertices.

Name="star([ 5 ]) in T^2 (VT)"
Dim=2

/SimplicialComplex], [SimplicialComplex

Properties known: Dim, FacetsEx, Name, Vertices.

Name="star([ 6 ]) in T^2 (VT)"
Dim=2

/SimplicialComplex], [SimplicialComplex

Properties known: Dim, FacetsEx, Name, Vertices.

Name="star([ 7 ]) in T^2 (VT)"
Dim=2

/SimplicialComplex] ]

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