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### 5 The GAP object types SCSimplicialComplex and SCNormalSurface

Currently, the GAP package simpcomp supports data structures for two different kinds of geometric objects, namely simplicial complexes (SCSimplicialComplex) and discrete normal surfaces (SCNormalSurface) which are both subtypes of the GAP object type SCPolyhedralComplex

#### 5.1 The object type SCSimplicialComplex

A major part of simpcomp deals with the object type SCSimplicialComplex. For a complete list of properties that SCSimplicialComplex handles, see Chapter 6. For a few fundamental methods and functions (such as checking the object class, copying objects of this type, etc.) for SCSimplicialComplex see below.

##### 5.1-1 SCIsSimplicialComplex
 ‣ SCIsSimplicialComplex( object ) ( filter )

Returns: true or false upon success, fail otherwise.

Checks if object is of type SCSimplicialComplex. The object type SCSimplicialComplex is derived from the object type SCPropertyObject.

 gap> c:=SCEmpty();;
gap> SCIsSimplicialComplex(c);
true


##### 5.1-2 SCCopy
 ‣ SCCopy( complex ) ( method )

Returns: a copy of complex upon success, fail otherwise.

Makes a deep copy'' of complex -- this is a copy such that all properties of the copy can be altered without changing the original complex.

 gap> c:=SCBdSimplex(4);;
gap> d:=SCCopy(c)-1;;
gap> c.Facets=d.Facets;
false

 gap> c:=SCBdSimplex(4);;
gap> d:=SCCopy(c);;
gap> IsIdenticalObj(c,d);
false


##### 5.1-3 ShallowCopy (SCSimplicialComplex)
 ‣ ShallowCopy (SCSimplicialComplex)( complex ) ( method )

Returns: a copy of complex upon success, fail otherwise.

Makes a copy of complex. This is actually a deep copy'' such that all properties of the copy can be altered without changing the original complex. Internally calls SCCopy (5.1-2).

 gap> c:=SCBdCrossPolytope(7);;
gap> d:=ShallowCopy(c)+10;;
gap> c.Facets=d.Facets;
false


##### 5.1-4 SCPropertiesDropped
 ‣ SCPropertiesDropped( complex ) ( function )

Returns: a object of type SCSimplicialComplex upon success, fail otherwise.

An object of the type SCSimplicialComplex caches its previously calculated properties such that each property only has to be calculated once. This function returns a copy of complex with all properties (apart from Facets, Dim and Name) dropped, clearing all previously computed properties. See also SCPropertyDrop (18.1-8) and SCPropertyTmpDrop (18.1-13).

 gap> c:=SC(SCFacets(SCBdCyclicPolytope(10,12)));
[SimplicialComplex

Properties known: Dim, FacetsEx, Name, Vertices.

Name="unnamed complex 27"
Dim=9

/SimplicialComplex]
gap> c.F; time;
[ 12, 66, 220, 495, 792, 922, 780, 465, 180, 36 ]
40
gap> c.F; time;
[ 12, 66, 220, 495, 792, 922, 780, 465, 180, 36 ]
0
gap> c:=SCPropertiesDropped(c);
[SimplicialComplex

Properties known: Dim, FacetsEx, Name, Vertices.

Name="unnamed complex 27"
Dim=9

/SimplicialComplex]
gap> c.F; time;
[ 12, 66, 220, 495, 792, 922, 780, 465, 180, 36 ]
36


#### 5.2 Overloaded operators of SCSimplicialComplex

simpcomp overloads some standard operations for the object type SCSimplicialComplex if this definition is intuitive and mathematically sound. See a list of overloaded operators below.

##### 5.2-1 Operation + (SCSimplicialComplex, Integer)
 ‣ Operation + (SCSimplicialComplex, Integer)( complex, value ) ( method )

Returns: the simplicial complex passed as argument upon success, fail otherwise.

Positively shifts the vertex labels of complex (provided that all labels satisfy the property IsAdditiveElement) by the amount specified in value.

 gap> c:=SCBdSimplex(3)+10;;
gap> c.Facets;
[ [ 11, 12, 13 ], [ 11, 12, 14 ], [ 11, 13, 14 ], [ 12, 13, 14 ] ]


##### 5.2-2 Operation - (SCSimplicialComplex, Integer)
 ‣ Operation - (SCSimplicialComplex, Integer)( complex, value ) ( method )

Returns: the simplicial complex passed as argument upon success, fail otherwise.

Negatively shifts the vertex labels of complex (provided that all labels satisfy the property IsAdditiveElement) by the amount specified in value.

 gap> c:=SCBdSimplex(3)-1;;
gap> c.Facets;
[ [ 0, 1, 2 ], [ 0, 1, 3 ], [ 0, 2, 3 ], [ 1, 2, 3 ] ]


##### 5.2-3 Operation mod (SCSimplicialComplex, Integer)
 ‣ Operation mod (SCSimplicialComplex, Integer)( complex, value ) ( method )

Returns: the simplicial complex passed as argument upon success, fail otherwise.

Takes all vertex labels of complex modulo the value specified in value (provided that all labels satisfy the property IsAdditiveElement). Warning: this might result in different vertices being assigned the same label or even in invalid facet lists, so be careful.

 gap> c:=(SCBdSimplex(3)*10) mod 7;;
gap> c.Facets;
[ [ 2, 3, 5 ], [ 2, 3, 6 ], [ 2, 5, 6 ], [ 3, 5, 6 ] ]


##### 5.2-4 Operation ^ (SCSimplicialComplex, Integer)
 ‣ Operation ^ (SCSimplicialComplex, Integer)( complex, value ) ( method )

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

Forms the value-th simplicial cartesian power of complex, i.e. the value-fold cartesian product of copies of complex. The complex passed as argument is not altered. Internally calls SCCartesianPower (6.6-1).

 gap> c:=SCBdSimplex(2)^2; #a torus
[SimplicialComplex

Properties known: Dim, FacetsEx, Name, TopologicalType, Vertices.

Name="(S^1_3)^2"
Dim=2
TopologicalType="(S^1)^2"

/SimplicialComplex]


##### 5.2-5 Operation + (SCSimplicialComplex, SCSimplicialComplex)
 ‣ Operation + (SCSimplicialComplex, SCSimplicialComplex)( complex1, complex2 ) ( method )

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

Forms the connected sum of complex1 and complex2. Uses the lexicographically first facets of both complexes to do the gluing. The complexes passed as arguments are not altered. Internally calls SCConnectedSum (6.6-5).

 gap> SCLib.SearchByName("RP^3");
[ [ 45, "RP^3" ], [ 111, "RP^3=L(2,1) (VT)" ], [ 589, "(S^2~S^1)#RP^3" ],
[ 590, "(S^2xS^1)#RP^3" ], [ 644, "(S^2~S^1)#2#RP^3" ],
[ 646, "(S^2xS^1)#2#RP^3" ], [ 2414, "RP^3#RP^3" ],
[ 2428, "RP^3=L(2,1) (VT)" ], [ 2488, "(S^2~S^1)#3#RP^3" ],
[ 2489, "(S^2xS^1)#3#RP^3" ], [ 2503, "RP^3=L(2,1) (VT)" ],
[ 7473, "(S^2xS^1)#4#RP^3" ], [ 7474, "(S^2~S^1)#4#RP^3" ],
[ 7505, "(S^2xS^1)#5#RP^3" ], [ 7507, "(S^2~S^1)#5#RP^3" ] ]
gap> SCLib.SearchByName("S^2~S^1"){[1..3]};
[ [ 12, "S^2~S^1 (VT)" ], [ 26, "S^2~S^1 (VT)" ], [ 30, "S^2~S^1 (VT)" ] ]
gap> c:=c+d; #form RP^3#(S^2~S^1)
[SimplicialComplex

Properties known: Dim, FacetsEx, Name, Vertices.

Name="RP^3#+-S^2~S^1 (VT)"
Dim=3

/SimplicialComplex]


##### 5.2-6 Operation - (SCSimplicialComplex, SCSimplicialComplex)
 ‣ Operation - (SCSimplicialComplex, SCSimplicialComplex)( complex1, complex2 ) ( method )

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

Calls SCDifference (6.10-5)(complex1, complex2)

##### 5.2-7 Operation * (SCSimplicialComplex, SCSimplicialComplex)
 ‣ Operation * (SCSimplicialComplex, SCSimplicialComplex)( complex1, complex2 ) ( method )

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

Forms the simplicial cartesian product of complex1 and complex2. Internally calls SCCartesianProduct (6.6-2).

 gap> SCLib.SearchByName("RP^2");
[ [ 3, "RP^2 (VT)" ], [ 645, "RP^2xS^1" ] ]
gap> c:=SCLib.Load(last[1][1])*SCBdSimplex(3); #form RP^2 x S^2
[SimplicialComplex

Properties known: Dim, FacetsEx, Name, Vertices.

Name="RP^2 (VT)xS^2_4"
Dim=4

/SimplicialComplex]


##### 5.2-8 Operation = (SCSimplicialComplex, SCSimplicialComplex)
 ‣ Operation = (SCSimplicialComplex, SCSimplicialComplex)( complex1, complex2 ) ( method )

Returns: true or false upon success, fail otherwise.

Calculates whether two simplicial complexes are isomorphic, i.e. are equal up to a relabeling of the vertices.

 gap> c:=SCBdSimplex(3);;
gap> c=c+10;
true
gap> c=SCBdCrossPolytope(4);
false


#### 5.3 SCSimplicialComplex as a subtype of Set

Apart from being a subtype of SCPropertyObject, an object of type SCSimplicialComplex also behaves like a GAP Set type. The elements of the set are given by the facets of the simplical complex, grouped by their dimensionality, i.e. if complex is an object of type SCSimplicialComplex, c[1] refers to the 0-faces of complex, c[2] to the 1-faces, etc.

##### 5.3-1 Operation Union (SCSimplicialComplex, SCSimplicialComplex)
 ‣ Operation Union (SCSimplicialComplex, SCSimplicialComplex)( complex1, complex2 ) ( method )

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

Computes the union of two simplicial complexes by calling SCUnion (7.3-16).

 gap> c:=Union(SCBdSimplex(3),SCBdSimplex(3)+3); #a wedge of two 2-spheres
[SimplicialComplex

Properties known: Dim, FacetsEx, Name, Vertices.

Name="S^2_4 cup S^2_4"
Dim=2

/SimplicialComplex]


##### 5.3-2 Operation Difference (SCSimplicialComplex, SCSimplicialComplex)
 ‣ Operation Difference (SCSimplicialComplex, SCSimplicialComplex)( complex1, complex2 ) ( method )

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

Computes the difference'' of two simplicial complexes by calling SCDifference (6.10-5).

 gap> c:=SCBdSimplex(3);;
gap> d:=SC([[1,2,3]]);;
gap> disc:=Difference(c,d);;
gap> disc.Facets;
[ [ 1, 2, 4 ], [ 1, 3, 4 ], [ 2, 3, 4 ] ]
gap> empty:=Difference(d,c);;
gap> empty.Dim;
-1


##### 5.3-3 Operation Intersection (SCSimplicialComplex, SCSimplicialComplex)
 ‣ Operation Intersection (SCSimplicialComplex, SCSimplicialComplex)( complex1, complex2 ) ( method )

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

Computes the intersection'' of two simplicial complexes by calling SCIntersection (6.10-8).

 gap> c:=SCBdSimplex(3);;
gap> d:=SCBdSimplex(3);;
gap> d:=SCMove(d,[[1,2,3],[]]);;
gap> d:=d+1;;
gap> s1:=SCIntersection(c,d);
[SimplicialComplex

Properties known: Dim, FacetsEx, Name, Vertices.

Name="S^2_4 cap unnamed complex 20"
Dim=1

/SimplicialComplex]
gap> s1.Facets;
[ [ 2, 3 ], [ 2, 4 ], [ 3, 4 ] ]


##### 5.3-4 Size (SCSimplicialComplex)
 ‣ Size (SCSimplicialComplex)( complex ) ( method )

Returns: an integer upon success, fail otherwise.

Returns the size'' of a simplicial complex. This is d+1, where d is the dimension of the complex. d+1 is returned instead of d, as all lists in GAP are indexed beginning with 1 -- thus this also holds for all the face lattice related properties of the complex.

 gap> SCLib.SearchByAttribute("F=[12,66,108,54]");
[ [ 139, "S^2xS^1 (VT)" ], [ 140, "S^2~S^1 (VT)" ], [ 141, "S^2xS^1 (VT)" ],
[ 142, "S^2xS^1 (VT)" ], [ 143, "(S^2xS^1)#(S^2xS^1) (VT)" ],
[ 144, "(S^2xS^1)#(S^2xS^1) (VT)" ], [ 145, "S^2xS^1 (VT)" ],
[ 146, "S^2xS^1 (VT)" ], [ 147, "S^2xS^1 (VT)" ], [ 148, "S^2~S^1 (VT)" ],
[ 149, "(S^2xS^1)#(S^2xS^1) (VT)" ], [ 150, "S^2~S^1 (VT)" ],
[ 151, "S^2~S^1 (VT)" ], [ 152, "S^2~S^1 (VT)" ], [ 153, "S^2~S^1 (VT)" ],
[ 154, "S^2~S^1 (VT)" ], [ 155, "S^2xS^1 (VT)" ], [ 156, "S^2xS^1 (VT)" ],
[ 157, "(S^2xS^1)#(S^2xS^1) (VT)" ], [ 158, "S^2~S^1 (VT)" ],
[ 159, "S^2xS^1 (VT)" ], [ 160, "(S^2xS^1)#(S^2xS^1) (VT)" ],
[ 161, "L_3_1" ] ]
gap> for i in [1..Size(c)] do Print(c.F[i],"\n"); od;
12
66
108
54


##### 5.3-5 Length (SCSimplicialComplex)
 ‣ Length (SCSimplicialComplex)( complex ) ( method )

Returns: an integer upon success, fail otherwise.

Returns the size'' of a simplicial complex by calling Size(complex).

 gap> SCLib.SearchByAttribute("F=[12,66,108,54]");
[ [ 139, "S^2xS^1 (VT)" ], [ 140, "S^2~S^1 (VT)" ], [ 141, "S^2xS^1 (VT)" ],
[ 142, "S^2xS^1 (VT)" ], [ 143, "(S^2xS^1)#(S^2xS^1) (VT)" ],
[ 144, "(S^2xS^1)#(S^2xS^1) (VT)" ], [ 145, "S^2xS^1 (VT)" ],
[ 146, "S^2xS^1 (VT)" ], [ 147, "S^2xS^1 (VT)" ], [ 148, "S^2~S^1 (VT)" ],
[ 149, "(S^2xS^1)#(S^2xS^1) (VT)" ], [ 150, "S^2~S^1 (VT)" ],
[ 151, "S^2~S^1 (VT)" ], [ 152, "S^2~S^1 (VT)" ], [ 153, "S^2~S^1 (VT)" ],
[ 154, "S^2~S^1 (VT)" ], [ 155, "S^2xS^1 (VT)" ], [ 156, "S^2xS^1 (VT)" ],
[ 157, "(S^2xS^1)#(S^2xS^1) (VT)" ], [ 158, "S^2~S^1 (VT)" ],
[ 159, "S^2xS^1 (VT)" ], [ 160, "(S^2xS^1)#(S^2xS^1) (VT)" ],
[ 161, "L_3_1" ] ]
gap> for i in [1..Length(c)] do Print(c.F[i],"\n"); od;
12
66
108
54


##### 5.3-6 Operation [] (SCSimplicialComplex)
 ‣ Operation [] (SCSimplicialComplex)( complex, pos ) ( method )

Returns: a list of faces upon success, fail otherwise.

Returns the (pos-1)-dimensional faces of complex as a list. If pos ≥ d+2, where d is the dimension of complex, the empty set is returned. Note that pos must be ≥ 1.

 gap> SCLib.SearchByName("K^2");
[ [ 17, "K^2 (VT)" ], [ 571, "K^2 (VT)" ] ]
gap> c[2];
[ [ 1, 2 ], [ 1, 3 ], [ 1, 5 ], [ 1, 7 ], [ 1, 9 ], [ 1, 10 ], [ 2, 3 ],
[ 2, 4 ], [ 2, 6 ], [ 2, 8 ], [ 2, 10 ], [ 3, 4 ], [ 3, 5 ], [ 3, 7 ],
[ 3, 9 ], [ 4, 5 ], [ 4, 6 ], [ 4, 8 ], [ 4, 10 ], [ 5, 6 ], [ 5, 7 ],
[ 5, 9 ], [ 6, 7 ], [ 6, 8 ], [ 6, 10 ], [ 7, 8 ], [ 7, 9 ], [ 8, 9 ],
[ 8, 10 ], [ 9, 10 ] ]
gap> c[4];
[  ]


##### 5.3-7 Iterator (SCSimplicialComplex)
 ‣ Iterator (SCSimplicialComplex)( complex ) ( method )

Returns: an iterator on the face lattice of complex upon success, fail otherwise.

Provides an iterator object for the face lattice of a simplicial complex.

 gap> c:=SCBdCrossPolytope(4);;
gap> for faces in c do Print(Length(faces),"\n"); od;
8
24
32
16


#### 5.4 The object type SCNormalSurface

The GAP object type SCNormalSurface is designed to describe slicings (level sets of discrete Morse functions) of combinatorial 3-manifolds, i. e. discrete normal surfaces. Internally SCNormalSurface is a subtype of SCPolyhedralComplex and, thus, mostly behaves like a SCSimplicialComplex object (see Section 5.1). For a very short introduction to normal surfaces see 2.4, for a more thorough introduction to the field see [Spr11b]. For some fundamental methods and functions for SCNormalSurface see below. For more functions related to the SCNormalSurface object type see Chapter 7.

#### 5.5 Overloaded operators of SCNormalSurface

As with the object type SCSimplicialComplex, simpcomp overloads some standard operations for the object type SCNormalSurface. See a list of overloaded operators below.

##### 5.5-1 Operation + (SCNormalSurface, Integer)
 ‣ Operation + (SCNormalSurface, Integer)( complex, value ) ( method )

Returns: the discrete normal surface passed as argument upon success, fail otherwise.

Positively shifts the vertex labels of complex (provided that all labels satisfy the property IsAdditiveElement) by the amount specified in value.

 gap> sl:=SCNSSlicing(SCBdSimplex(4),[[1],[2..5]]);;
gap> sl.Facets;
[ [ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ] ], [ [ 1, 2 ], [ 1, 3 ], [ 1, 5 ] ],
[ [ 1, 2 ], [ 1, 4 ], [ 1, 5 ] ], [ [ 1, 3 ], [ 1, 4 ], [ 1, 5 ] ] ]
gap> sl:=sl + 2;;
gap> sl.Facets;
[ [ [ 3, 4 ], [ 3, 5 ], [ 3, 6 ] ], [ [ 3, 4 ], [ 3, 5 ], [ 3, 7 ] ],
[ [ 3, 4 ], [ 3, 6 ], [ 3, 7 ] ], [ [ 3, 5 ], [ 3, 6 ], [ 3, 7 ] ] ]


##### 5.5-2 Operation - (SCNormalSurface, Integer)
 ‣ Operation - (SCNormalSurface, Integer)( complex, value ) ( method )

Returns: the discrete normal surface passed as argument upon success, fail otherwise.

Negatively shifts the vertex labels of complex (provided that all labels satisfy the property IsAdditiveElement) by the amount specified in value.

 gap> sl:=SCNSSlicing(SCBdSimplex(4),[[1],[2..5]]);;
gap> sl.Facets;
[ [ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ] ], [ [ 1, 2 ], [ 1, 3 ], [ 1, 5 ] ],
[ [ 1, 2 ], [ 1, 4 ], [ 1, 5 ] ], [ [ 1, 3 ], [ 1, 4 ], [ 1, 5 ] ] ]
gap> sl:=sl - 2;;
gap> sl.Facets;
[ [ [ -1, 0 ], [ -1, 1 ], [ -1, 2 ] ], [ [ -1, 0 ], [ -1, 1 ], [ -1, 3 ] ],
[ [ -1, 0 ], [ -1, 2 ], [ -1, 3 ] ], [ [ -1, 1 ], [ -1, 2 ], [ -1, 3 ] ] ]


##### 5.5-3 Operation mod (SCNormalSurface, Integer)
 ‣ Operation mod (SCNormalSurface, Integer)( complex, value ) ( method )

Returns: the discrete normal surface passed as argument upon success, fail otherwise.

Takes all vertex labels of complex modulo the value specified in value (provided that all labels satisfy the property IsAdditiveElement). Warning: this might result in different vertices being assigned the same label or even invalid facet lists, so be careful.

 gap> sl:=SCNSSlicing(SCBdSimplex(4),[[1],[2..5]]);;
gap> sl.Facets;
[ [ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ] ], [ [ 1, 2 ], [ 1, 3 ], [ 1, 5 ] ],
[ [ 1, 2 ], [ 1, 4 ], [ 1, 5 ] ], [ [ 1, 3 ], [ 1, 4 ], [ 1, 5 ] ] ]
gap> sl:=sl mod 2;;
gap> sl.Facets;
[ [ [ 1, 0 ], [ 1, 0 ], [ 1, 1 ] ], [ [ 1, 0 ], [ 1, 0 ], [ 1, 1 ] ],
[ [ 1, 0 ], [ 1, 1 ], [ 1, 1 ] ], [ [ 1, 0 ], [ 1, 1 ], [ 1, 1 ] ] ]


#### 5.6 SCNormalSurface as a subtype of Set

Like objects of type SCSimplicialComplex, an object of type SCNormalSurface behaves like a GAP Set type. The elements of the set are given by the facets of the normal surface, grouped by their dimensionality and type, i.e. if complex is an object of type SCNormalSurface, c[1] refers to the 0-faces of complex, c[2] to the 1-faces, c[3] to the triangles and c[4] to the quadrilaterals. See below for some examples and Section 5.3 for details.

##### 5.6-1 Operation Union (SCNormalSurface, SCNormalSurface)
 ‣ Operation Union (SCNormalSurface, SCNormalSurface)( complex1, complex2 ) ( method )

Returns: discrete normal surface of type SCNormalSurface upon success, fail otherwise.

Computes the union of two discrete normal surfaces by calling SCUnion (7.3-16).

 gap> SCLib.SearchByAttribute("F = [ 10, 35, 50, 25 ]");
[ [ 19, "S^3 (VT)" ] ]