  
  [1X4 [33X[0;0YFunctions and operations for the GAP object type [10XSCPolyhedralComplex[110X[101X[1X[133X[101X
  
  [33X[0;0YIn  the following all operations for the [5XGAP[105X object type [10XSCPolyhedralComplex[110X
  are  listed.  I.  e.  for  the  following  operations  only  one  method  is
  implemented  to  deal  with  all  geometric objects derived from this object
  type.[133X
  
  
  [1X4.1 [33X[0;0YComputing properties of objects of type [10XSCPolyhedralComplex[110X[101X[1X[133X[101X
  
  [33X[0;0YThe  following  functions  compute  basic  properties  of  objects  of  type
  [10XSCPolyhedralComplex[110X  (and  thus  also of objects of type [10XSCSimplicialComplex[110X
  and  [10XSCNormalSurface[110X).  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 [10XShallowCopy[110X or the
  internal [5Xsimpcomp[105X function [10XSCIntFunc.DeepCopy[110X to get a mutable copy.[133X
  
  [33X[0;0YNote:  every  object  is internally stored with the standard vertex labeling
  from [22X1[122X to [22Xn[122X 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.[133X
  
  [1X4.1-1 SCFacets[101X
  
  [29X[2XSCFacets[102X( [3Xcomplex[103X ) [32X method
  [6XReturns:[106X  [33X[0;10Ya facet list upon success, [9Xfail[109X otherwise.[133X
  
  [33X[0;0YReturns the facets of a simplicial complex in the original vertex labeling.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X gap> c:=SC([[2,3],[3,4],[4,2]]);;[128X[104X
    [4X[28X gap> SCFacets(c);[128X[104X
    [4X[28X [ [ 2, 3 ], [ 2, 4 ], [ 3, 4 ] ][128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [1X4.1-2 SCFacetsEx[101X
  
  [29X[2XSCFacetsEx[102X( [3Xcomplex[103X ) [32X method
  [6XReturns:[106X  [33X[0;10Ya facet list upon success, [9Xfail[109X otherwise.[133X
  
  [33X[0;0YReturns  the  facets  of a simplicial complex as they are stored, i. e. with
  standard vertex labeling from 1 to n.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X gap> c:=SC([[2,3],[3,4],[4,2]]);;[128X[104X
    [4X[28X gap> SCFacetsEx(c);[128X[104X
    [4X[28X [ [ 1, 2 ], [ 1, 3 ], [ 2, 3 ] ][128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [1X4.1-3 SCVertices[101X
  
  [29X[2XSCVertices[102X( [3Xcomplex[103X ) [32X method
  [6XReturns:[106X  [33X[0;10Ya list of vertex labels of [3Xcomplex[103X upon success, [9Xfail[109X otherwise.[133X
  
  [33X[0;0YReturns the vertex labels of a simplicial complex [3Xcomplex[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X gap> sphere:=SC([["x",45,[1,1]],["x",45,["b",3]],["x",[1,1],[128X[104X
    [4X[28X   ["b",3]],[45,[1,1],["b",3]]]);;[128X[104X
    [4X[28X gap> SCVerticesEx(sphere);[128X[104X
    [4X[28X [ 1 .. 4 ][128X[104X
    [4X[28X gap> SCVertices(sphere);[128X[104X
    [4X[28X [ 45, [ 1, 1 ], "x", [ "b", 3 ] ][128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [1X4.1-4 SCVerticesEx[101X
  
  [29X[2XSCVerticesEx[102X( [3Xcomplex[103X ) [32X method
  [6XReturns:[106X  [33X[0;10Y[22X[ 1, ... , n ][122X upon success, [9Xfail[109X otherwise.[133X
  
  [33X[0;0YReturns  [22X[1,  ...  ,  n ][122X, where [22Xn[122X is the number of vertices of a simplicial
  complex [3Xcomplex[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X gap> c:=SC([[1,4,5],[4,9,8],[12,13,14,15,16,17]]);;[128X[104X
    [4X[28X gap> SCVerticesEx(c);[128X[104X
    [4X[28X [ 1 .. 11 ][128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  
  [1X4.2 [33X[0;0YVertex labelings and label operations[133X[101X
  
  [33X[0;0YThis  section focuses on functions operating on the labels of a complex such
  as the name or the vertex labeling.[133X
  
  [33X[0;0YInternally,  [5Xsimpcomp[105X  uses  the  standard  labeling  [22X[1,  ...  ,  n][122X. It is
  recommended  to  use  simple  vertex  labels  like  integers  and,  whenever
  possible, the standard labeling, see also [2XSCRelabelStandard[102X ([14X4.2-7[114X).[133X
  
  [1X4.2-1 SCLabelMax[101X
  
  [29X[2XSCLabelMax[102X( [3Xcomplex[103X ) [32X method
  [6XReturns:[106X  [33X[0;10Yvertex  label of [3Xcomplex[103X (an integer, a short list, a character, a
            short string) upon success, [9Xfail[109X otherwise.[133X
  
  [33X[0;0YThe  maximum  over  all  vertex  labels  is  determined  by the [5XGAP[105X function
  [10XMaximumList[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X gap> c:=SCBdSimplex(3);;
[128X[104X
    [4X[28X gap> SCRelabel(c,[10,100,100000,3500]);;
[128X[104X
    [4X[28X gap> SCLabelMax(c);
[128X[104X
    [4X[28X 100000[128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [4X[32X  Example  [32X[104X
    [4X[28X gap> c:=SCBdSimplex(3);;
[128X[104X
    [4X[28X gap> SCRelabel(c,["a","bbb",5,[1,1]]);;
[128X[104X
    [4X[28X gap> SCLabelMax(c);
[128X[104X
    [4X[28X "bbb"[128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [1X4.2-2 SCLabelMin[101X
  
  [29X[2XSCLabelMin[102X( [3Xcomplex[103X ) [32X method
  [6XReturns:[106X  [33X[0;10Yvertex  label of [3Xcomplex[103X (an integer, a short list, a character, a
            short string) upon success, [9Xfail[109X otherwise.[133X
  
  [33X[0;0YThe  minimum  over  all  vertex  labels  is  determined  by the [5XGAP[105X function
  [10XMinimumList[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X gap> c:=SCBdSimplex(3);;
[128X[104X
    [4X[28X gap> SCRelabel(c,[10,100,100000,3500]);;
[128X[104X
    [4X[28X gap> SCLabelMin(c);
[128X[104X
    [4X[28X 10[128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [4X[32X  Example  [32X[104X
    [4X[28X gap> c:=SCBdSimplex(3);;
[128X[104X
    [4X[28X gap> SCRelabel(c,["a","bbb",5,[1,1]]);;
[128X[104X
    [4X[28X gap> SCLabelMin(c);
[128X[104X
    [4X[28X 5[128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [1X4.2-3 SCLabels[101X
  
  [29X[2XSCLabels[102X( [3Xcomplex[103X ) [32X method
  [6XReturns:[106X  [33X[0;10Ya  list  of  vertex  labels  of [3Xcomplex[103X (a list of integers, short
            lists,   characters,   short  strings,  ...)  upon  success,  [9Xfail[109X
            otherwise.[133X
  
  [33X[0;0YReturns  the  vertex  labels  of  [3Xcomplex[103X  as  a  list. This is a synonym of
  [2XSCVertices[102X ([14X4.1-3[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X gap> c:=SCFromFacets(Combinations(["a","b","c","d"],3));;
[128X[104X
    [4X[28X gap> SCLabels(c);
[128X[104X
    [4X[28X [ "a", "b", "c", "d" ][128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [1X4.2-4 SCName[101X
  
  [29X[2XSCName[102X( [3Xcomplex[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Ya string upon success, [9Xfail[109X otherwise.[133X
  
  [33X[0;0YReturns the name of a simplicial complex [3Xcomplex[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X gap> c:=SCBdSimplex(5);;
[128X[104X
    [4X[28X gap> SCName(c);
[128X[104X
    [4X[28X "S^4_6"[128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [4X[32X  Example  [32X[104X
    [4X[28X gap> c:=SC([[1,2],[2,3],[3,1]]);;
[128X[104X
    [4X[28X gap> SCName(c);
[128X[104X
    [4X[28X "unnamed complex 2"[128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [1X4.2-5 SCReference[101X
  
  [29X[2XSCReference[102X( [3Xcomplex[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Ya string upon success, [9Xfail[109X otherwise.[133X
  
  [33X[0;0YReturns a literature reference of a polyhedral complex [3Xcomplex[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X gap> c:=SCLib.Load(253);;
[128X[104X
    [4X[28X gap> SCReference(c);
[128X[104X
    [4X[28X "F.H.Lutz: 'The Manifold Page', http://www.math.tu-berlin.de/diskregeom/stella\[128X[104X
    [4X[28X r/"[128X[104X
    [4X[28X gap> c:=SC([[1,2],[2,3],[3,1]]);;
[128X[104X
    [4X[28X gap> SCReference(c);
[128X[104X
    [4X[28X #I  SCReference: complex lacks reference.[128X[104X
    [4X[28X fail[128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [1X4.2-6 SCRelabel[101X
  
  [29X[2XSCRelabel[102X( [3Xcomplex[103X, [3Xmaptable[103X ) [32X method
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X upon success, [9Xfail[109X otherwise.[133X
  
  [33X[0;0Y[3Xmaptable[103X  has  to be a list of length [22Xn[122X where [22Xn[122X is the number of vertices of
  [3Xcomplex[103X.  The  function maps the [22Xi[122X-th entry of [3Xmaptable[103X to the [22Xi[122X-th entry of
  the  current  vertex labels. If [3Xcomplex[103X has the standard vertex labeling [22X[1,
  ... , n][122X the vertex label [22Xi[122X is mapped to [3Xmaptable[i][103X.[133X
  
  [33X[0;0YNote  that  the  elements  of  [3Xmaptable[103X  must admit a total ordering. Hence,
  following Section 4.11 of the [5XGAP[105X manual, they must be members of one of the
  following  families: rationals [10XIsRat[110X, cyclotomics [10XIsCyclotomic[110X, finite field
  elements  [10XIsFFE[110X, permutations [10XIsPerm[110X, booleans [10XIsBool[110X, characters [10XIsChar[110X and
  lists (strings) [10XIsList[110X.[133X
  
  [33X[0;0YInternally the property ``SCVertices'' of [3Xcomplex[103X is replaced by [3Xmaptable.[103X[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X gap> list:=SCLib.SearchByAttribute("F[1]=12");; 
[128X[104X
    [4X[28X gap> c:=SCLib.Load(list[1][1]);;
[128X[104X
    [4X[28X gap> SCVertices(c);
[128X[104X
    [4X[28X [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ][128X[104X
    [4X[28X gap> SCRelabel(c,["a","b","c","d","e","f","g","h","i","j","k","l"]);
[128X[104X
    [4X[28X true[128X[104X
    [4X[28X gap> SCLabels(c);
[128X[104X
    [4X[28X [ "a", "b", "c", "d", "e", "f", "g", "h", "i", "j", "k", "l" ][128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [1X4.2-7 SCRelabelStandard[101X
  
  [29X[2XSCRelabelStandard[102X( [3Xcomplex[103X ) [32X method
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X upon success, [9Xfail[109X otherwise.[133X
  
  [33X[0;0YMaps  vertex  labels [22Xv_1 , ... , v_n[122X of [3Xcomplex[103X to [22X[1 , ... , n][122X. Internally
  the property "SCVertices" is replaced by [22X[1 , ... , n][122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X gap> list:=SCLib.SearchByAttribute("F[1]=12");; 
[128X[104X
    [4X[28X gap> c:=SCLib.Load(list[1][1]);;
[128X[104X
    [4X[28X gap> SCRelabel(c,[4..15]);
[128X[104X
    [4X[28X true[128X[104X
    [4X[28X gap> SCVertices(c);
[128X[104X
    [4X[28X [ 4 .. 15 ][128X[104X
    [4X[28X gap> SCRelabelStandard(c);
[128X[104X
    [4X[28X true[128X[104X
    [4X[28X gap> SCLabels(c);
[128X[104X
    [4X[28X [ 1 .. 12 ][128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [1X4.2-8 SCRelabelTransposition[101X
  
  [29X[2XSCRelabelTransposition[102X( [3Xcomplex[103X, [3Xpair[103X ) [32X method
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X upon success, [9Xfail[109X otherwise.[133X
  
  [33X[0;0YPermutes  vertex  labels of a single pair of vertices. [3Xpair[103X has to be a list
  of length [22X2[122X and a sublist of the property ``SCVertices''.[133X
  
  [33X[0;0YThe   function  is  equivalent  to  [2XSCRelabel[102X  ([14X4.2-6[114X)  with  [3Xmaptable[103X  [22X=  [
  SCVertices[1]  ,  ...  ,  SCVertices[j]  ,  ...  ,  SCVertices[i]  ,  dots ,
  SCVertices[n]][122X if [3Xpair[103X [22X= [ SCVertices[j] , SCVertices[i]][122X, [22Xj ≤ i[122X, [22Xj ≠ i[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X gap> c:=SCBdSimplex(3);;
[128X[104X
    [4X[28X gap> SCVertices(c);
[128X[104X
    [4X[28X [ 1, 2, 3, 4 ][128X[104X
    [4X[28X gap> SCRelabelTransposition(c,[1,2]);;
[128X[104X
    [4X[28X gap> SCLabels(c);
[128X[104X
    [4X[28X [ 2, 1, 3, 4 ][128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [1X4.2-9 SCRename[101X
  
  [29X[2XSCRename[102X( [3Xcomplex[103X, [3Xname[103X ) [32X method
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X upon success, [9Xfail[109X otherwise.[133X
  
  [33X[0;0YRenames a polyhedral complex. The argument [3Xname[103X has to be given in form of a
  string.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X gap> c:=SCBdSimplex(5);;
[128X[104X
    [4X[28X gap> SCName(c);
[128X[104X
    [4X[28X "S^4_6"[128X[104X
    [4X[28X gap> SCRename(c,"mySphere");
[128X[104X
    [4X[28X true[128X[104X
    [4X[28X gap> SCName(c);
[128X[104X
    [4X[28X "mySphere"[128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [1X4.2-10 SCSetReference[101X
  
  [29X[2XSCSetReference[102X( [3Xcomplex[103X, [3Xref[103X ) [32X method
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X upon success, [9Xfail[109X otherwise.[133X
  
  [33X[0;0YSets  the literature reference of a polyhedral complex. The argument [3Xref[103X has
  to be given in form of a string.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X gap> c:=SCBdSimplex(5);;
[128X[104X
    [4X[28X gap> SCReference(c);
[128X[104X
    [4X[28X #I  SCReference: complex lacks reference.[128X[104X
    [4X[28X fail[128X[104X
    [4X[28X gap> SCSetReference(c,"my 5-sphere in my cool paper");
[128X[104X
    [4X[28X true[128X[104X
    [4X[28X gap> SCReference(c);
[128X[104X
    [4X[28X "my 5-sphere in my cool paper"[128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [1X4.2-11 SCUnlabelFace[101X
  
  [29X[2XSCUnlabelFace[102X( [3Xcomplex[103X, [3Xface[103X ) [32X method
  [6XReturns:[106X  [33X[0;10Ya list upon success, [9Xfail[109X otherwise.[133X
  
  [33X[0;0YComputes the standard labeling of [3Xface[103X in [3Xcomplex[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X gap> c:=SCBdSimplex(3);;
[128X[104X
    [4X[28X gap> SCRelabel(c,["a","bbb",5,[1,1]]);;
[128X[104X
    [4X[28X gap> SCUnlabelFace(c,["a","bbb",5]);
[128X[104X
    [4X[28X [ 1, 2, 3 ][128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  
  [1X4.3 [33X[0;0YOperations on objects of type [10XSCPolyhedralComplex[110X[101X[1X[133X[101X
  
  [33X[0;0YThe   following   functions   perform   operations   on   objects   of  type
  [10XSCPolyhedralComplex[110X  and all of its subtypes. Most of them return simplicial
  complexes.  Thus,  this  section is closely related to the Sections [14X6.6[114X (for
  objects  of  type [10XSCSimplicialComplex[110X), ''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.[133X
  
  [1X4.3-1 SCAntiStar[101X
  
  [29X[2XSCAntiStar[102X( [3Xcomplex[103X, [3Xface[103X ) [32X method
  [6XReturns:[106X  [33X[0;10Ysimplicial  complex of type [10XSCSimplicialComplex[110X upon success, [9Xfail[109X
            otherwise .[133X
  
  [33X[0;0YComputes  the  anti  star  of  [3Xface[103X (a face given as a list of vertices or a
  scalar  interpreted  as  vertex) in [3Xcomplex[103X, i. e. the complement of [3Xface[103X in
  [3Xcomplex[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X gap> SCLib.SearchByName("RP^2");     
[128X[104X
    [4X[28X [ [ 3, "RP^2 (VT)" ], [ 635, "RP^2xS^1" ] ][128X[104X
    [4X[28X gap> rp2:=SCLib.Load(last[1][1]);;
[128X[104X
    [4X[28X gap> SCVertices(rp2);
[128X[104X
    [4X[28X [ 1, 2, 3, 4, 5, 6 ][128X[104X
    [4X[28X gap> SCAntiStar(rp2,1);
[128X[104X
    [4X[28X [SimplicialComplex[128X[104X
    [4X[28X [128X[104X
    [4X[28X  Properties known: Dim, FacetsEx, Name, Vertices.[128X[104X
    [4X[28X [128X[104X
    [4X[28X  Name="ast([ 1 ]) in RP^2 (VT)"[128X[104X
    [4X[28X  Dim=2[128X[104X
    [4X[28X [128X[104X
    [4X[28X /SimplicialComplex][128X[104X
    [4X[28X gap> last.Facets;
[128X[104X
    [4X[28X [ [ 2, 3, 4 ], [ 2, 4, 5 ], [ 2, 5, 6 ], [ 3, 4, 6 ], [ 3, 5, 6 ] ][128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [1X4.3-2 SCLink[101X
  
  [29X[2XSCLink[102X( [3Xcomplex[103X, [3Xface[103X ) [32X method
  [6XReturns:[106X  [33X[0;10Ysimplicial  complex of type [10XSCSimplicialComplex[110X upon success, [9Xfail[109X
            otherwise.[133X
  
  [33X[0;0YComputes  the  link  of [3Xface[103X (a face given as a list of vertices or a scalar
  interpreted  as  vertex)  in  a polyhedral complex [3Xcomplex[103X, i. e. all facets
  containing  [3Xface[103X, reduced by [3Xface[103X. if [3Xcomplex[103X is pure, the resulting complex
  is  of  dimension  dim([3Xcomplex[103X)  -  dim([3Xface[103X)  [22X-1[122X.  If [3Xface[103X is not a face of
  [3Xcomplex[103X the empty complex is returned.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X gap> SCLib.SearchByName("RP^2");     
[128X[104X
    [4X[28X [ [ 3, "RP^2 (VT)" ], [ 635, "RP^2xS^1" ] ][128X[104X
    [4X[28X gap> rp2:=SCLib.Load(last[1][1]);;
[128X[104X
    [4X[28X gap> SCVertices(rp2);
[128X[104X
    [4X[28X [ 1, 2, 3, 4, 5, 6 ][128X[104X
    [4X[28X gap> SCLink(rp2,[1]);
[128X[104X
    [4X[28X [SimplicialComplex[128X[104X
    [4X[28X [128X[104X
    [4X[28X  Properties known: Dim, FacetsEx, Name, Vertices.[128X[104X
    [4X[28X [128X[104X
    [4X[28X  Name="lk([ 1 ]) in RP^2 (VT)"[128X[104X
    [4X[28X  Dim=1[128X[104X
    [4X[28X [128X[104X
    [4X[28X /SimplicialComplex][128X[104X
    [4X[28X gap> last.Facets;
[128X[104X
    [4X[28X [ [ 2, 3 ], [ 2, 6 ], [ 3, 5 ], [ 4, 5 ], [ 4, 6 ] ][128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [1X4.3-3 SCLinks[101X
  
  [29X[2XSCLinks[102X( [3Xcomplex[103X, [3Xk[103X ) [32X method
  [6XReturns:[106X  [33X[0;10Ya  list  of  simplicial complexes of type [10XSCSimplicialComplex[110X upon
            success, [9Xfail[109X otherwise.[133X
  
  [33X[0;0YComputes  the  link  of  all  [3Xk[103X-faces  of the polyhedral complex [3Xcomplex[103X and
  returns  them  as  a  list  of simplicial complexes. Internally calls [2XSCLink[102X
  ([14X4.3-2[114X) for every [3Xk[103X-face of [3Xcomplex[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X gap> c:=SCBdSimplex(4);;
[128X[104X
    [4X[28X gap> SCLinks(c,0);
[128X[104X
    [4X[28X [ [SimplicialComplex[128X[104X
    [4X[28X     [128X[104X
    [4X[28X      Properties known: Dim, FacetsEx, Name, Vertices.[128X[104X
    [4X[28X     [128X[104X
    [4X[28X      Name="lk([ 1 ]) in S^3_5"[128X[104X
    [4X[28X      Dim=2[128X[104X
    [4X[28X     [128X[104X
    [4X[28X     /SimplicialComplex], [SimplicialComplex[128X[104X
    [4X[28X     [128X[104X
    [4X[28X      Properties known: Dim, FacetsEx, Name, Vertices.[128X[104X
    [4X[28X     [128X[104X
    [4X[28X      Name="lk([ 2 ]) in S^3_5"[128X[104X
    [4X[28X      Dim=2[128X[104X
    [4X[28X     [128X[104X
    [4X[28X     /SimplicialComplex], [SimplicialComplex[128X[104X
    [4X[28X     [128X[104X
    [4X[28X      Properties known: Dim, FacetsEx, Name, Vertices.[128X[104X
    [4X[28X     [128X[104X
    [4X[28X      Name="lk([ 3 ]) in S^3_5"[128X[104X
    [4X[28X      Dim=2[128X[104X
    [4X[28X     [128X[104X
    [4X[28X     /SimplicialComplex], [SimplicialComplex[128X[104X
    [4X[28X     [128X[104X
    [4X[28X      Properties known: Dim, FacetsEx, Name, Vertices.[128X[104X
    [4X[28X     [128X[104X
    [4X[28X      Name="lk([ 4 ]) in S^3_5"[128X[104X
    [4X[28X      Dim=2[128X[104X
    [4X[28X     [128X[104X
    [4X[28X     /SimplicialComplex], [SimplicialComplex[128X[104X
    [4X[28X     [128X[104X
    [4X[28X      Properties known: Dim, FacetsEx, Name, Vertices.[128X[104X
    [4X[28X     [128X[104X
    [4X[28X      Name="lk([ 5 ]) in S^3_5"[128X[104X
    [4X[28X      Dim=2[128X[104X
    [4X[28X     [128X[104X
    [4X[28X     /SimplicialComplex] ][128X[104X
    [4X[28X gap> SCLinks(c,1);
[128X[104X
    [4X[28X [ [SimplicialComplex[128X[104X
    [4X[28X     [128X[104X
    [4X[28X      Properties known: Dim, FacetsEx, Name, Vertices.[128X[104X
    [4X[28X     [128X[104X
    [4X[28X      Name="lk([ 1, 2 ]) in S^3_5"[128X[104X
    [4X[28X      Dim=1[128X[104X
    [4X[28X     [128X[104X
    [4X[28X     /SimplicialComplex], [SimplicialComplex[128X[104X
    [4X[28X     [128X[104X
    [4X[28X      Properties known: Dim, FacetsEx, Name, Vertices.[128X[104X
    [4X[28X     [128X[104X
    [4X[28X      Name="lk([ 1, 3 ]) in S^3_5"[128X[104X
    [4X[28X      Dim=1[128X[104X
    [4X[28X     [128X[104X
    [4X[28X     /SimplicialComplex], [SimplicialComplex[128X[104X
    [4X[28X     [128X[104X
    [4X[28X      Properties known: Dim, FacetsEx, Name, Vertices.[128X[104X
    [4X[28X     [128X[104X
    [4X[28X      Name="lk([ 1, 4 ]) in S^3_5"[128X[104X
    [4X[28X      Dim=1[128X[104X
    [4X[28X     [128X[104X
    [4X[28X     /SimplicialComplex], [SimplicialComplex[128X[104X
    [4X[28X     [128X[104X
    [4X[28X      Properties known: Dim, FacetsEx, Name, Vertices.[128X[104X
    [4X[28X     [128X[104X
    [4X[28X      Name="lk([ 1, 5 ]) in S^3_5"[128X[104X
    [4X[28X      Dim=1[128X[104X
    [4X[28X     [128X[104X
    [4X[28X     /SimplicialComplex], [SimplicialComplex[128X[104X
    [4X[28X     [128X[104X
    [4X[28X      Properties known: Dim, FacetsEx, Name, Vertices.[128X[104X
    [4X[28X     [128X[104X
    [4X[28X      Name="lk([ 2, 3 ]) in S^3_5"[128X[104X
    [4X[28X      Dim=1[128X[104X
    [4X[28X     [128X[104X
    [4X[28X     /SimplicialComplex], [SimplicialComplex[128X[104X
    [4X[28X     [128X[104X
    [4X[28X      Properties known: Dim, FacetsEx, Name, Vertices.[128X[104X
    [4X[28X     [128X[104X
    [4X[28X      Name="lk([ 2, 4 ]) in S^3_5"[128X[104X
    [4X[28X      Dim=1[128X[104X
    [4X[28X     [128X[104X
    [4X[28X     /SimplicialComplex], [SimplicialComplex[128X[104X
    [4X[28X     [128X[104X
    [4X[28X      Properties known: Dim, FacetsEx, Name, Vertices.[128X[104X
    [4X[28X     [128X[104X
    [4X[28X      Name="lk([ 2, 5 ]) in S^3_5"[128X[104X
    [4X[28X      Dim=1[128X[104X
    [4X[28X     [128X[104X
    [4X[28X     /SimplicialComplex], [SimplicialComplex[128X[104X
    [4X[28X     [128X[104X
    [4X[28X      Properties known: Dim, FacetsEx, Name, Vertices.[128X[104X
    [4X[28X     [128X[104X
    [4X[28X      Name="lk([ 3, 4 ]) in S^3_5"[128X[104X
    [4X[28X      Dim=1[128X[104X
    [4X[28X     [128X[104X
    [4X[28X     /SimplicialComplex], [SimplicialComplex[128X[104X
    [4X[28X     [128X[104X
    [4X[28X      Properties known: Dim, FacetsEx, Name, Vertices.[128X[104X
    [4X[28X     [128X[104X
    [4X[28X      Name="lk([ 3, 5 ]) in S^3_5"[128X[104X
    [4X[28X      Dim=1[128X[104X
    [4X[28X     [128X[104X
    [4X[28X     /SimplicialComplex], [SimplicialComplex[128X[104X
    [4X[28X     [128X[104X
    [4X[28X      Properties known: Dim, FacetsEx, Name, Vertices.[128X[104X
    [4X[28X     [128X[104X
    [4X[28X      Name="lk([ 4, 5 ]) in S^3_5"[128X[104X
    [4X[28X      Dim=1[128X[104X
    [4X[28X     [128X[104X
    [4X[28X     /SimplicialComplex] ][128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [1X4.3-4 SCStar[101X
  
  [29X[2XSCStar[102X( [3Xcomplex[103X, [3Xface[103X ) [32X method
  [6XReturns:[106X  [33X[0;10Ysimplicial  complex of type [10XSCSimplicialComplex[110X upon success, [9Xfail[109X
            otherwise .[133X
  
  [33X[0;0YComputes  the  star  of [3Xface[103X (a face given as a list of vertices or a scalar
  interpreted  as  vertex)  in  a polyhedral complex [3Xcomplex[103X, i. e. the set of
  facets of [3Xcomplex[103X that contain [3Xface[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X gap> SCLib.SearchByName("RP^2");     
[128X[104X
    [4X[28X [ [ 3, "RP^2 (VT)" ], [ 635, "RP^2xS^1" ] ][128X[104X
    [4X[28X gap> rp2:=SCLib.Load(last[1][1]);;
[128X[104X
    [4X[28X gap> SCVertices(rp2);
[128X[104X
    [4X[28X [ 1, 2, 3, 4, 5, 6 ][128X[104X
    [4X[28X gap> SCStar(rp2,1);
[128X[104X
    [4X[28X [SimplicialComplex[128X[104X
    [4X[28X [128X[104X
    [4X[28X  Properties known: Dim, FacetsEx, Name, Vertices.[128X[104X
    [4X[28X [128X[104X
    [4X[28X  Name="star([ 1 ]) in RP^2 (VT)"[128X[104X
    [4X[28X  Dim=2[128X[104X
    [4X[28X [128X[104X
    [4X[28X /SimplicialComplex][128X[104X
    [4X[28X gap> last.Facets;
[128X[104X
    [4X[28X [ [ 1, 2, 3 ], [ 1, 2, 6 ], [ 1, 3, 5 ], [ 1, 4, 5 ], [ 1, 4, 6 ] ][128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [1X4.3-5 SCStars[101X
  
  [29X[2XSCStars[102X( [3Xcomplex[103X, [3Xk[103X ) [32X method
  [6XReturns:[106X  [33X[0;10Ya  list  of  simplicial complexes of type [10XSCSimplicialComplex[110X upon
            success, [9Xfail[109X otherwise.[133X
  
  [33X[0;0YComputes  the  star  of  all  [3Xk[103X-faces  of the polyhedral complex [3Xcomplex[103X and
  returns  them  as  a  list  of simplicial complexes. Internally calls [2XSCStar[102X
  ([14X4.3-4[114X) for every [3Xk[103X-face of [3Xcomplex[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X gap> SCLib.SearchByName("T^2"){[1..6]};
[128X[104X
    [4X[28X [ [ 4, "T^2 (VT)" ], [ 5, "T^2 (VT)" ], [ 9, "T^2 (VT)" ], [ 10, "T^2 (VT)" ],[128X[104X
    [4X[28X   [ 18, "T^2 (VT)" ], [ 20, "(T^2)#2" ] ][128X[104X
    [4X[28X gap> torus:=SCLib.Load(last[1][1]);; # the minimal 7-vertex torus
[128X[104X
    [4X[28X gap> SCStars(torus,0); # 7 2-discs as vertex stars
[128X[104X
    [4X[28X [ [SimplicialComplex[128X[104X
    [4X[28X     [128X[104X
    [4X[28X      Properties known: Dim, FacetsEx, Name, Vertices.[128X[104X
    [4X[28X     [128X[104X
    [4X[28X      Name="star([ 1 ]) in T^2 (VT)"[128X[104X
    [4X[28X      Dim=2[128X[104X
    [4X[28X     [128X[104X
    [4X[28X     /SimplicialComplex], [SimplicialComplex[128X[104X
    [4X[28X     [128X[104X
    [4X[28X      Properties known: Dim, FacetsEx, Name, Vertices.[128X[104X
    [4X[28X     [128X[104X
    [4X[28X      Name="star([ 2 ]) in T^2 (VT)"[128X[104X
    [4X[28X      Dim=2[128X[104X
    [4X[28X     [128X[104X
    [4X[28X     /SimplicialComplex], [SimplicialComplex[128X[104X
    [4X[28X     [128X[104X
    [4X[28X      Properties known: Dim, FacetsEx, Name, Vertices.[128X[104X
    [4X[28X     [128X[104X
    [4X[28X      Name="star([ 3 ]) in T^2 (VT)"[128X[104X
    [4X[28X      Dim=2[128X[104X
    [4X[28X     [128X[104X
    [4X[28X     /SimplicialComplex], [SimplicialComplex[128X[104X
    [4X[28X     [128X[104X
    [4X[28X      Properties known: Dim, FacetsEx, Name, Vertices.[128X[104X
    [4X[28X     [128X[104X
    [4X[28X      Name="star([ 4 ]) in T^2 (VT)"[128X[104X
    [4X[28X      Dim=2[128X[104X
    [4X[28X     [128X[104X
    [4X[28X     /SimplicialComplex], [SimplicialComplex[128X[104X
    [4X[28X     [128X[104X
    [4X[28X      Properties known: Dim, FacetsEx, Name, Vertices.[128X[104X
    [4X[28X     [128X[104X
    [4X[28X      Name="star([ 5 ]) in T^2 (VT)"[128X[104X
    [4X[28X      Dim=2[128X[104X
    [4X[28X     [128X[104X
    [4X[28X     /SimplicialComplex], [SimplicialComplex[128X[104X
    [4X[28X     [128X[104X
    [4X[28X      Properties known: Dim, FacetsEx, Name, Vertices.[128X[104X
    [4X[28X     [128X[104X
    [4X[28X      Name="star([ 6 ]) in T^2 (VT)"[128X[104X
    [4X[28X      Dim=2[128X[104X
    [4X[28X     [128X[104X
    [4X[28X     /SimplicialComplex], [SimplicialComplex[128X[104X
    [4X[28X     [128X[104X
    [4X[28X      Properties known: Dim, FacetsEx, Name, Vertices.[128X[104X
    [4X[28X     [128X[104X
    [4X[28X      Name="star([ 7 ]) in T^2 (VT)"[128X[104X
    [4X[28X      Dim=2[128X[104X
    [4X[28X     [128X[104X
    [4X[28X     /SimplicialComplex] ][128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
