chap7.txt

  
  7 Functions and operations for SCNormalSurface
  
  
  7.1 Creating an SCNormalSurface object
  
  This  section  contains functions to construct discrete normal surfaces that
  are   slicings   from   a   list  of  2-dimensional  facets  (triangles  and
  quadrilaterals) or combinatorial 3-manifolds.
  
  For  a very short introduction to the theory of discrete normal surfaces and
  slicings  see  Section  2.4  and Section 2.5, for an introduction to the GAP
  object  type  SCNormalSurface  see 5.4, for more information see the article
  [Spr11b].
  
  7.1-1 SCNSEmpty
  
  SCNSEmpty(  )  function
  Returns:  discrete normal surface of type SCNormalSurface upon success, fail
            otherwise.
  
  Generates  an  empty  complex  (of  dimension  -1),  i. e. an object of type
  SCNormalSurface with empty facet list.
  
    Example  
     gap> SCNSEmpty();
     [NormalSurface
     
      Properties known: Dim, FacetsEx, Name, Vertices.
     
      Name="empty normal surface"
      Dim=-1
     
     /NormalSurface]
     
  
  
  7.1-2 SCNSFromFacets
  
  SCNSFromFacets( facets )  method
  Returns:  discrete normal surface of type SCNormalSurface upon success, fail
            otherwise.
  
  Constructor   for   a  discrete  normal  surface  from  a  facet  list,  see
  SCFromFacets (6.1-1) for details.
  
    Example  
     gap> sl:=SCNSFromFacets([[1,2,3],[1,2,4,5],[1,3,4,6],[2,3,5,6],[4,5,6]]);
     [NormalSurface
     
      Properties known: Dim, FacetsEx, Name, Vertices.
     
      Name="unnamed complex 86"
      Dim=2
     
     /NormalSurface]
     
  
  
  7.1-3 SCNS
  
  SCNS( facets )  method
  Returns:  discrete normal surface of type SCNormalSurface upon success, fail
            otherwise.
  
  Internally calls SCNSFromFacets (7.1-2).
  
    Example  
     gap> sl:=SCNS([[1,2,3],[1,2,4,5],[1,3,4,6],[2,3,5,6],[4,5,6]]);
     [NormalSurface
     
      Properties known: Dim, FacetsEx, Name, Vertices.
     
      Name="unnamed complex 87"
      Dim=2
     
     /NormalSurface]
     
  
  
  7.1-4 SCNSSlicing
  
  SCNSSlicing( complex, slicing )  function
  Returns:  discrete normal surface of type SCNormalSurface upon success, fail
            otherwise.
  
  Computes  a slicing defined by a partition slicing of the set of vertices of
  the  3-dimensional  combinatorial  pseudomanifold  complex.  In  particular,
  slicing  has  to  be a pair of lists of vertex labels and has to contain all
  vertex labels of complex.
  
    Example  
     gap> SCLib.SearchByAttribute("F=[ 10, 35, 50, 25 ]");
     [ [ 19, "S^3 (VT)" ] ]
     gap> c:=SCLib.Load(last[1][1]);;                       
     gap> sl:=SCNSSlicing(c,[[1..5],[6..10]]);    
     [NormalSurface
     
      Properties known: ConnectedComponents, Dim, EulerCharacteristic, FVector, Fac\
     etsEx, Genus, IsConnected, IsOrientable, NSTriangulation, Name, TopologicalTyp\
     e, Vertices.
     
      Name="slicing [ [ 1, 2, 3, 4, 5 ], [ 6, 7, 8, 9, 10 ] ] of S^3 (VT)"
      Dim=2
      FVector=[ 17, 36, 12, 9 ]
      EulerCharacteristic=2
      IsOrientable=true
      TopologicalType="S^2"
     
     /NormalSurface]
     gap> sl.Facets;
     [ [ [ 1, 6 ], [ 1, 8 ], [ 1, 9 ] ], [ [ 1, 6 ], [ 1, 8 ], [ 3, 6 ], [ 3, 8 ] ]
         , [ [ 1, 6 ], [ 1, 9 ], [ 4, 6 ], [ 4, 9 ] ], 
       [ [ 1, 6 ], [ 3, 6 ], [ 4, 6 ] ], [ [ 1, 8 ], [ 1, 9 ], [ 1, 10 ] ], 
       [ [ 1, 8 ], [ 1, 10 ], [ 3, 8 ], [ 3, 10 ] ], 
       [ [ 1, 9 ], [ 1, 10 ], [ 2, 9 ], [ 2, 10 ] ], 
       [ [ 1, 9 ], [ 2, 9 ], [ 4, 9 ] ], [ [ 1, 10 ], [ 2, 10 ], [ 3, 10 ] ], 
       [ [ 2, 7 ], [ 2, 9 ], [ 2, 10 ] ], 
       [ [ 2, 7 ], [ 2, 9 ], [ 4, 7 ], [ 4, 9 ] ], 
       [ [ 2, 7 ], [ 2, 10 ], [ 5, 7 ], [ 5, 10 ] ], 
       [ [ 2, 7 ], [ 4, 7 ], [ 5, 7 ] ], [ [ 2, 10 ], [ 3, 10 ], [ 5, 10 ] ], 
       [ [ 3, 6 ], [ 3, 8 ], [ 5, 6 ], [ 5, 8 ] ], [ [ 3, 6 ], [ 4, 6 ], [ 5, 6 ] ]
         , [ [ 3, 8 ], [ 3, 10 ], [ 5, 8 ], [ 5, 10 ] ], 
       [ [ 4, 6 ], [ 4, 7 ], [ 4, 9 ] ], [ [ 4, 6 ], [ 4, 7 ], [ 5, 6 ], [ 5, 7 ] ]
         , [ [ 5, 6 ], [ 5, 7 ], [ 5, 8 ] ], [ [ 5, 7 ], [ 5, 8 ], [ 5, 10 ] ] ]
     gap> sl:=SCNSSlicing(c,[[1,3,5,7,9],[2,4,6,8,10]]);    
     [NormalSurface
     
      Properties known: ConnectedComponents, Dim, EulerCharacteristic, FVector, Fac\
     etsEx, Genus, IsConnected, IsOrientable, NSTriangulation, Name, TopologicalTyp\
     e, Vertices.
     
      Name="slicing [ [ 1, 3, 5, 7, 9 ], [ 2, 4, 6, 8, 10 ] ] of S^3 (VT)"
      Dim=2
      FVector=[ 25, 50, 0, 25 ]
      EulerCharacteristic=0
      IsOrientable=true
      TopologicalType="T^2"
     
     /NormalSurface]
     gap> sl.Facets;                           
     [ [ [ 1, 2 ], [ 1, 4 ], [ 3, 2 ], [ 3, 4 ] ], 
       [ [ 1, 2 ], [ 1, 4 ], [ 9, 2 ], [ 9, 4 ] ], 
       [ [ 1, 2 ], [ 1, 10 ], [ 3, 2 ], [ 3, 10 ] ], 
       [ [ 1, 2 ], [ 1, 10 ], [ 9, 2 ], [ 9, 10 ] ], 
       [ [ 1, 4 ], [ 1, 6 ], [ 3, 4 ], [ 3, 6 ] ], 
       [ [ 1, 4 ], [ 1, 6 ], [ 9, 4 ], [ 9, 6 ] ], 
       [ [ 1, 6 ], [ 1, 8 ], [ 3, 6 ], [ 3, 8 ] ], 
       [ [ 1, 6 ], [ 1, 8 ], [ 9, 6 ], [ 9, 8 ] ], 
       [ [ 1, 8 ], [ 1, 10 ], [ 3, 8 ], [ 3, 10 ] ], 
       [ [ 1, 8 ], [ 1, 10 ], [ 9, 8 ], [ 9, 10 ] ], 
       [ [ 3, 2 ], [ 3, 4 ], [ 5, 2 ], [ 5, 4 ] ], 
       [ [ 3, 2 ], [ 3, 10 ], [ 5, 2 ], [ 5, 10 ] ], 
       [ [ 3, 4 ], [ 3, 6 ], [ 5, 4 ], [ 5, 6 ] ], 
       [ [ 3, 6 ], [ 3, 8 ], [ 5, 6 ], [ 5, 8 ] ], 
       [ [ 3, 8 ], [ 3, 10 ], [ 5, 8 ], [ 5, 10 ] ], 
       [ [ 5, 2 ], [ 5, 4 ], [ 7, 2 ], [ 7, 4 ] ], 
       [ [ 5, 2 ], [ 5, 10 ], [ 7, 2 ], [ 7, 10 ] ], 
       [ [ 5, 4 ], [ 5, 6 ], [ 7, 4 ], [ 7, 6 ] ], 
       [ [ 5, 6 ], [ 5, 8 ], [ 7, 6 ], [ 7, 8 ] ], 
       [ [ 5, 8 ], [ 5, 10 ], [ 7, 8 ], [ 7, 10 ] ], 
       [ [ 7, 2 ], [ 7, 4 ], [ 9, 2 ], [ 9, 4 ] ], 
       [ [ 7, 2 ], [ 7, 10 ], [ 9, 2 ], [ 9, 10 ] ], 
       [ [ 7, 4 ], [ 7, 6 ], [ 9, 4 ], [ 9, 6 ] ], 
       [ [ 7, 6 ], [ 7, 8 ], [ 9, 6 ], [ 9, 8 ] ], 
       [ [ 7, 8 ], [ 7, 10 ], [ 9, 8 ], [ 9, 10 ] ] ]
     
  
  
  
  7.2 Generating new objects from discrete normal surfaces
  
  simpcomp  provides  the  possibility  to  copy  and  / or triangulate normal
  surfaces.  Note  that  other  constructions  like  the  connected sum or the
  cartesian  product  do  not  make  sense  for  (embedded) normal surfaces in
  general.
  
  7.2-1 SCCopy
  
  SCCopy( complex )  method
  Returns:  discrete normal surface of type SCNormalSurface upon success, fail
            otherwise.
  
  Copies a GAP object of type SCNormalSurface (cf. SCCopy).
  
    Example  
     gap> sl:=SCNSSlicing(SCBdSimplex(4),[[1],[2..5]]);
     [NormalSurface
     
      Properties known: ConnectedComponents, Dim, EulerCharacteristic, FVector, Fac\
     etsEx, Genus, IsConnected, IsOrientable, NSTriangulation, Name, TopologicalTyp\
     e, Vertices.
     
      Name="slicing [ [ 1 ], [ 2, 3, 4, 5 ] ] of S^3_5"
      Dim=2
      FVector=[ 4, 6, 4 ]
      EulerCharacteristic=2
      IsOrientable=true
      TopologicalType="S^2"
     
     /NormalSurface]
     gap> sl_2:=SCCopy(sl);                          
     [NormalSurface
     
      Properties known: ConnectedComponents, Dim, EulerCharacteristic, FVector, Fac\
     etsEx, Genus, IsConnected, IsOrientable, NSTriangulation, Name, TopologicalTyp\
     e, Vertices.
     
      Name="slicing [ [ 1 ], [ 2, 3, 4, 5 ] ] of S^3_5"
      Dim=2
      FVector=[ 4, 6, 4 ]
      EulerCharacteristic=2
      IsOrientable=true
      TopologicalType="S^2"
     
     /NormalSurface]
     gap> IsIdenticalObj(sl,sl_2);                     
     false
     
  
  
  7.2-2 SCNSTriangulation
  
  SCNSTriangulation( sl )  method
  Returns:  simplicial  complex of type SCSimplicialComplex upon success, fail
            otherwise.
  
  Computes  a  simplicial  subdivision of a slicing sl without introducing new
  vertices. The subdivision is stored as a property of sl and thus is returned
  as  an  immutable  object.  Note  that  symmetry  may  be  lost  during  the
  computation.
  
    Example  
     gap> SCLib.SearchByAttribute("F=[ 10, 35, 50, 25 ]");
     [ [ 19, "S^3 (VT)" ] ]
     gap> c:=SCLib.Load(last[1][1]);;
     gap> sl:=SCNSSlicing(c,[[1,3,5,7,9],[2,4,6,8,10]]);;
     gap> sl.F; 
     [ 25, 50, 0, 25 ]
     gap> sc:=SCNSTriangulation(sl);;
     gap> sc.F;
     [ 25, 75, 50 ]
     
  
  
  
  7.3 Properties of SCNormalSurface objects
  
  Although  some  properties  of  a discrete normal surface can be computed by
  using  the  functions  for  simplicial  complexes,  there  is  a  variety of
  properties needing specially designed functions. See below for a list.
  
  7.3-1 SCConnectedComponents
  
  SCConnectedComponents( complex )  method
  Returns:  a  list  of  simplicial  complexes  of  type  SCNormalSurface upon
            success, fail otherwise.
  
  Computes all connected components of an arbitrary normal surface.
  
    Example  
     gap> sl:=SCNSSlicing(SCBdCrossPolytope(4),[[1,2],[3..8]]);
     [NormalSurface
     
      Properties known: ConnectedComponents, Dim, EulerCharacteristic, FVector, Fac\
     etsEx, IsConnected, IsOrientable, NSTriangulation, Name, TopologicalType, Vert\
     ices.
     
      Name="slicing [ [ 1, 2 ], [ 3, 4, 5, 6, 7, 8 ] ] of Bd(\beta^4)"
      Dim=2
      FVector=[ 12, 24, 16 ]
      EulerCharacteristic=4
      IsOrientable=true
      TopologicalType="S^2 U S^2"
     
     /NormalSurface]
     gap> cc:=SCConnectedComponents(sl);
     [ [NormalSurface
         
          Properties known: Dim, EulerCharacteristic, FVector, FacetsEx, Genus, IsC\
     onnected, IsOrientable, NSTriangulation, Name, TopologicalType, Vertices.
         
          Name="unnamed complex 302_cc_#1"
          Dim=2
          FVector=[ 6, 12, 8 ]
          EulerCharacteristic=2
          IsOrientable=true
          TopologicalType="S^2"
         
         /NormalSurface], [NormalSurface
         
          Properties known: Dim, EulerCharacteristic, FVector, FacetsEx, Genus, IsC\
     onnected, IsOrientable, NSTriangulation, Name, TopologicalType, Vertices.
         
          Name="unnamed complex 302_cc_#2"
          Dim=2
          FVector=[ 6, 12, 8 ]
          EulerCharacteristic=2
          IsOrientable=true
          TopologicalType="S^2"
         
         /NormalSurface] ]
     
  
  
  7.3-2 SCDim
  
  SCDim( sl )  method
  Returns:  an integer upon success, fail otherwise.
  
  Computes  the  dimension  of a discrete normal surface (which is always 2 if
  the slicing sl is not empty).
  
    Example  
     gap> sl:=SCNSEmpty();;                                                    
     gap> SCDim(sl);                                                         
     -1
     gap> sl:=SCNSFromFacets([[1,2,3],[1,2,4,5],[1,3,4,6],[2,3,5,6],[4,5,6]]);;
     gap> SCDim(sl);                                                         
     2
     
  
  
  7.3-3 SCEulerCharacteristic
  
  SCEulerCharacteristic( sl )  method
  Returns:  an integer upon success, fail otherwise.
  
  Computes  the  Euler  characteristic  of  a  discrete normal surface sl, cf.
  SCEulerCharacteristic.
  
    Example  
     gap> list:=SCLib.SearchByName("S^2xS^1");;  
     gap> c:=SCLib.Load(list[1][1]);;             
     gap> sl:=SCNSSlicing(c,[[1..5],[6..10]]);;
     gap> SCEulerCharacteristic(sl);                 
     4
     
  
  
  7.3-4 SCFVector
  
  SCFVector( sl )  method
  Returns:  a  1,  3 or 4 tuple of (non-negative) integer values upon success,
            fail otherwise.
  
  Computes  the  f-vector  of  a  discrete normal surface, i. e. the number of
  vertices, edges, triangles and quadrilaterals of sl, cf. SCFVector.
  
    Example  
     gap> list:=SCLib.SearchByName("S^2xS^1");;
     gap> c:=SCLib.Load(list[1][1]);;             
     gap> sl:=SCNSSlicing(c,[[1..5],[6..10]]);;
     gap> SCFVector(sl);                 
     [ 20, 40, 16, 8 ]
     
  
  
  7.3-5 SCFaceLattice
  
  SCFaceLattice( complex )  method
  Returns:  a list of facet lists upon success, fail otherwise.
  
  Computes  the  face  lattice of a discrete normal surface sl in the original
  labeling.  Triangles  and  quadrilaterals  are stored separately (cf. SCSkel
  (6.9-54)).
  
    Example  
     gap> c:=SCBdSimplex(4);;              
     gap> sl:=SCNSSlicing(c,[[1,2],[3..5]]);;
     gap> SCFaceLattice(sl);                            
     [ [ [ [ 1, 3 ] ], [ [ 1, 4 ] ], [ [ 1, 5 ] ], [ [ 2, 3 ] ], [ [ 2, 4 ] ], 
           [ [ 2, 5 ] ] ], 
       [ [ [ 1, 3 ], [ 1, 4 ] ], [ [ 1, 3 ], [ 1, 5 ] ], [ [ 1, 3 ], [ 2, 3 ] ], 
           [ [ 1, 4 ], [ 1, 5 ] ], [ [ 1, 4 ], [ 2, 4 ] ], [ [ 1, 5 ], [ 2, 5 ] ], 
           [ [ 2, 3 ], [ 2, 4 ] ], [ [ 2, 3 ], [ 2, 5 ] ], [ [ 2, 4 ], [ 2, 5 ] ] ]
         , [ [ [ 1, 3 ], [ 1, 4 ], [ 1, 5 ] ], [ [ 2, 3 ], [ 2, 4 ], [ 2, 5 ] ] ], 
       [ [ [ 1, 3 ], [ 1, 4 ], [ 2, 3 ], [ 2, 4 ] ], 
           [ [ 1, 3 ], [ 1, 5 ], [ 2, 3 ], [ 2, 5 ] ], 
           [ [ 1, 4 ], [ 1, 5 ], [ 2, 4 ], [ 2, 5 ] ] ] ]
     gap> sl.F;
     [ 6, 9, 2, 3 ]
     
  
  
  7.3-6 SCFaceLatticeEx
  
  SCFaceLatticeEx( complex )  method
  Returns:  a list of face lists upon success, fail otherwise.
  
  Computes  the  face  lattice of a discrete normal surface sl in the standard
  labeling.  Triangles  and quadrilaterals are stored separately (cf. SCSkelEx
  (6.9-55)).
  
    Example  
     gap> c:=SCBdSimplex(4);;              
     gap> sl:=SCNSSlicing(c,[[1,2],[3..5]]);;
     gap> SCFaceLatticeEx(sl);                            
     [ [ [ 1 ], [ 2 ], [ 3 ], [ 4 ], [ 5 ], [ 6 ] ], 
       [ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 2, 3 ], [ 2, 5 ], [ 3, 6 ], [ 4, 5 ], 
           [ 4, 6 ], [ 5, 6 ] ], [ [ 1, 2, 3 ], [ 4, 5, 6 ] ], 
       [ [ 1, 2, 4, 5 ], [ 1, 3, 4, 6 ], [ 2, 3, 5, 6 ] ] ]
     gap> sl.F;
     [ 6, 9, 2, 3 ]
     
  
  
  7.3-7 SCFpBettiNumbers
  
  SCFpBettiNumbers( sl, p )  method
  Returns:  a list of non-negative integers upon success, fail otherwise.
  
  Computes  the  Betti  numbers  modulo  p  of a slicing sl. Internally, sl is
  triangulated  (using  SCNSTriangulation  (7.2-2))  and the Betti numbers are
  computed via SCFpBettiNumbers using the triangulation.
  
    Example  
     gap> SCLib.SearchByName("(S^2xS^1)#20");       
     [ [ 7617, "(S^2xS^1)#20" ] ]
     gap> c:=SCLib.Load(last[1][1]);;
     gap> c.F;
     [ 27, 298, 542, 271 ]
     gap> sl:=SCNSSlicing(c,[[1..13],[14..27]]);;
     gap> SCFpBettiNumbers(sl,2);
     [ 2, 14, 2 ]
     
  
  
  7.3-8 SCGenus
  
  SCGenus( sl )  method
  Returns:  a non-negative integer upon success, fail otherwise.
  
  Computes the genus of a discrete normal surface sl.
  
    Example  
     gap> SCLib.SearchByName("(S^2xS^1)#20");
     [ [ 7617, "(S^2xS^1)#20" ] ]
     gap> c:=SCLib.Load(last[1][1]);;               
     gap> c.F;                               
     [ 27, 298, 542, 271 ]
     gap> sl:=SCNSSlicing(c,[[1..12],[13..27]]);;
     gap> SCIsConnected(sl);
     true
     gap> SCGenus(sl);                     
     7
     
  
  
  7.3-9 SCHomology
  
  SCHomology( sl )  method
  Returns:  a list of homology groups upon success, fail otherwise.
  
  Computes  the  homology of a slicing sl. Internally, sl is triangulated (cf.
  SCNSTriangulation   (7.2-2))   and   simplicial  homology  is  computed  via
  SCHomology using the triangulation.
  
    Example  
     gap> SCLib.SearchByName("(S^2xS^1)#20");       
     [ [ 7617, "(S^2xS^1)#20" ] ]
     gap> c:=SCLib.Load(last[1][1]);;
     gap> c.F;
     [ 27, 298, 542, 271 ]
     gap> sl:=SCNSSlicing(c,[[1..12],[13..27]]);;   
     gap> sl.Homology;
     [ [ 0, [  ] ], [ 14, [  ] ], [ 1, [  ] ] ]
     gap> sl:=SCNSSlicing(c,[[1..13],[14..27]]);;
     gap> sl.Homology;                       
     [ [ 1, [  ] ], [ 14, [  ] ], [ 2, [  ] ] ]
     
  
  
  7.3-10 SCIsConnected
  
  SCIsConnected( complex )  method
  Returns:  true or false upon success, fail otherwise.
  
  Checks if a normal surface complex is connected.
  
    Example  
     gap> list:=SCLib.SearchByAttribute("Dim=3 and F[1]=10");;
     gap> c:=SCLib.Load(list[1][1]);
     [SimplicialComplex
     
      Properties known: AltshulerSteinberg, AutomorphismGroup, 
                        AutomorphismGroupSize, AutomorphismGroupStructure, 
                        AutomorphismGroupTransitivity, ConnectedComponents, 
                        Dim, DualGraph, EulerCharacteristic, FVector, 
                        FacetsEx, GVector, GeneratorsEx, HVector, 
                        HasBoundary, HasInterior, Homology, Interior, 
                        IsCentrallySymmetric, IsConnected, 
                        IsEulerianManifold, IsManifold, IsOrientable, 
                        IsPseudoManifold, IsPure, IsStronglyConnected, 
                        MinimalNonFacesEx, Name, Neighborliness, 
                        NumFaces[], Orientation, Reference, SkelExs[], 
                        Vertices.
     
      Name="S^3 (VT)"
      Dim=3
      AltshulerSteinberg=0
      AutomorphismGroupSize=200
      AutomorphismGroupStructure="(D10 x D10) : C2"
      AutomorphismGroupTransitivity=1
      EulerCharacteristic=0
      FVector=[ 10, 35, 50, 25 ]
      GVector=[ 5, 5 ]
      HVector=[ 6, 11, 6, 1 ]
      HasBoundary=false
      HasInterior=true
      Homology=[ [ 0, [ ] ], [ 0, [ ] ], [ 0, [ ] ], [ 1, [ ] ] ]
      IsCentrallySymmetric=false
      IsConnected=true
      IsEulerianManifold=true
      IsOrientable=true
      IsPseudoManifold=true
      IsPure=true
      IsStronglyConnected=true
      Neighborliness=1
     
     /SimplicialComplex]
     gap> sl:=SCNSSlicing(c,[[1..5],[6..10]]);
     [NormalSurface
     
      Properties known: ConnectedComponents, Dim, EulerCharacteristic, FVector, Fac\
     etsEx, Genus, IsConnected, IsOrientable, NSTriangulation, Name, TopologicalTyp\
     e, Vertices.
     
      Name="slicing [ [ 1, 2, 3, 4, 5 ], [ 6, 7, 8, 9, 10 ] ] of S^3 (VT)"
      Dim=2
      FVector=[ 17, 36, 12, 9 ]
      EulerCharacteristic=2
      IsOrientable=true
      TopologicalType="S^2"
     
     /NormalSurface]
     gap> SCIsConnected(sl);
     true
     
  
  
  7.3-11 SCIsEmpty
  
  SCIsEmpty( complex )  method
  Returns:  true or false upon success, fail otherwise.
  
  Checks  if  a  normal  surface  complex  is  the  empty  complex,  i.  e.  a
  SCNormalSurface object with empty facet list.
  
    Example  
     gap> sl:=SCNS([]);;
     gap> SCIsEmpty(sl);
     true
     
  
  
  7.3-12 SCIsOrientable
  
  SCIsOrientable( sl )  method
  Returns:  true or false upon success, fail otherwise.
  
  Checks if a discrete normal surface sl is orientable.
  
    Example  
     gap> c:=SCBdSimplex(4);;
     gap> sl:=SCNSSlicing(c,[[1,2],[3,4,5]]);
     [NormalSurface
     
      Properties known: ConnectedComponents, Dim, EulerCharacteristic, FVector, Fac\
     etsEx, Genus, IsConnected, IsOrientable, NSTriangulation, Name, TopologicalTyp\
     e, Vertices.
     
      Name="slicing [ [ 1, 2 ], [ 3, 4, 5 ] ] of S^3_5"
      Dim=2
      FVector=[ 6, 9, 2, 3 ]
      EulerCharacteristic=2
      IsOrientable=true
      TopologicalType="S^2"
     
     /NormalSurface]
     gap> SCIsOrientable(sl);
     true
     
  
  
  7.3-13 SCSkel
  
  SCSkel( sl, k )  method
  Returns:  a  face  list (of k+1tuples) or a list of face lists upon success,
            fail otherwise.
  
  Computes  all  faces  of  cardinality  k+1  in  the original labeling: k = 0
  computes  the  vertices,  k  =  1  computes  the  edges,  k = 2 computes the
  triangles, k = 3 computes the quadrilaterals.
  
  If  k  is  a  list  (necessarily  a sublist of [ 0,1,2,3 ]) all faces of all
  cardinalities contained in k are computed.
  
    Example  
     gap> c:=SCBdSimplex(4);;              
     gap> sl:=SCNSSlicing(c,[[1],[2..5]]);;
     gap> SCSkel(sl,1);                            
     [ [ [ 1, 2 ], [ 1, 3 ] ], [ [ 1, 2 ], [ 1, 4 ] ], [ [ 1, 2 ], [ 1, 5 ] ], 
       [ [ 1, 3 ], [ 1, 4 ] ], [ [ 1, 3 ], [ 1, 5 ] ], [ [ 1, 4 ], [ 1, 5 ] ] ]
     
  
  
    Example  
     gap> c:=SCBdSimplex(4);;              
     gap> sl:=SCNSSlicing(c,[[1],[2..5]]);;
     gap> SCSkel(sl,3);                            
     [  ]
     gap> sl:=SCNSSlicing(c,[[1,2],[3..5]]);;
     gap> SCSkelEx(sl,3);                            
     [ [ 1, 2, 4, 5 ], [ 1, 3, 4, 6 ], [ 2, 3, 5, 6 ] ]
     
  
  
  7.3-14 SCSkelEx
  
  SCSkelEx( sl, k )  method
  Returns:  a  face  list (of k+1tuples) or a list of face lists upon success,
            fail otherwise.
  
  Computes  all  faces  of  cardinality  k+1  in  the standard labeling: k = 0
  computes  the  vertices,  k  =  1  computes  the  edges,  k = 2 computes the
  triangles, k = 3 computes the quadrilaterals.
  
  If  k  is  a  list  (necessarily  a sublist of [ 0,1,2,3 ]) all faces of all
  cardinalities contained in k are computed.
  
    Example  
     gap> c:=SCBdSimplex(4);;              
     gap> sl:=SCNSSlicing(c,[[1],[2..5]]);;
     gap> SCSkelEx(sl,1);                            
     [ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 2, 3 ], [ 2, 4 ], [ 3, 4 ] ]
     
  
  
    Example  
     gap> c:=SCBdSimplex(4);;              
     gap> sl:=SCNSSlicing(c,[[1],[2..5]]);;
     gap> SCSkelEx(sl,3);                            
     [  ]
     gap> sl:=SCNSSlicing(c,[[1,2],[3..5]]);;
     gap> SCSkelEx(sl,3);                            
     [ [ 1, 2, 4, 5 ], [ 1, 3, 4, 6 ], [ 2, 3, 5, 6 ] ]
     
  
  
  7.3-15 SCTopologicalType
  
  SCTopologicalType( sl )  method
  Returns:  a string upon success, fail otherwise.
  
  Determines  the  topological  type  of sl via the classification theorem for
  closed  compact  surfaces.  If  sl is not connected, the topological type of
  each connected component is computed.
  
    Example  
     gap> SCLib.SearchByName("(S^2xS^1)#20");      
     [ [ 7617, "(S^2xS^1)#20" ] ]
     gap> c:=SCLib.Load(last[1][1]);;
     gap> c.F;
     [ 27, 298, 542, 271 ]
     gap> for i in [1..26] do sl:=SCNSSlicing(c,[[1..i],[i+1..27]]); Print(sl.TopologicalType,"\n"); od;                                           
     S^2
     S^2
     S^2
     S^2
     S^2 U S^2
     S^2 U S^2
     S^2
     (T^2)#3
     (T^2)#5
     (T^2)#4
     (T^2)#3
     (T^2)#7
     (T^2)#7 U S^2
     (T^2)#7 U S^2
     (T^2)#7 U S^2
     (T^2)#8 U S^2
     (T^2)#7 U S^2
     (T^2)#8
     (T^2)#6
     (T^2)#6
     (T^2)#5
     (T^2)#3
     (T^2)#2
     T^2
     S^2
     S^2
     
  
  
  7.3-16 SCUnion
  
  SCUnion( complex1, complex2 )  method
  Returns:  normal   surface   of  type  SCNormalSurface  upon  success,  fail
            otherwise.
  
  Forms  the  union of two normal surfaces complex1 and complex2 as the normal
  surface  formed  by the union of their facet sets. The two arguments are not
  altered.  Note:  for the union process the vertex labelings of the complexes
  are   taken   into  account,  see  also  Operation  Union  (SCNormalSurface,
  SCNormalSurface)  (5.6-1). Facets occurring in both arguments are treated as
  one facet in the new complex.
  
    Example  
     gap> list:=SCLib.SearchByAttribute("Dim=3 and F[1]=10");;
     gap> c:=SCLib.Load(list[1][1]);
     [SimplicialComplex
     
      Properties known: AltshulerSteinberg, AutomorphismGroup, 
                        AutomorphismGroupSize, AutomorphismGroupStructure, 
                        AutomorphismGroupTransitivity, ConnectedComponents, 
                        Dim, DualGraph, EulerCharacteristic, FVector, 
                        FacetsEx, GVector, GeneratorsEx, HVector, 
                        HasBoundary, HasInterior, Homology, Interior, 
                        IsCentrallySymmetric, IsConnected, 
                        IsEulerianManifold, IsManifold, IsOrientable, 
                        IsPseudoManifold, IsPure, IsStronglyConnected, 
                        MinimalNonFacesEx, Name, Neighborliness, 
                        NumFaces[], Orientation, Reference, SkelExs[], 
                        Vertices.
     
      Name="S^3 (VT)"
      Dim=3
      AltshulerSteinberg=0
      AutomorphismGroupSize=200
      AutomorphismGroupStructure="(D10 x D10) : C2"
      AutomorphismGroupTransitivity=1
      EulerCharacteristic=0
      FVector=[ 10, 35, 50, 25 ]
      GVector=[ 5, 5 ]
      HVector=[ 6, 11, 6, 1 ]
      HasBoundary=false
      HasInterior=true
      Homology=[ [ 0, [ ] ], [ 0, [ ] ], [ 0, [ ] ], [ 1, [ ] ] ]
      IsCentrallySymmetric=false
      IsConnected=true
      IsEulerianManifold=true
      IsOrientable=true
      IsPseudoManifold=true
      IsPure=true
      IsStronglyConnected=true
      Neighborliness=1
     
     /SimplicialComplex]
     gap> sl1:=SCNSSlicing(c,[[1..5],[6..10]]);;
     gap> sl2:=sl1+10;;
     gap> sl3:=SCUnion(sl1,sl2);;
     gap> SCTopologicalType(sl3);
     "S^2 U S^2"