5 The GAP object types

`SCSimplicialComplex`

and `SCNormalSurface`

5.2 Overloaded operators of

5.2-1 Operation + (SCSimplicialComplex, Integer)

5.2-2 Operation - (SCSimplicialComplex, Integer)

5.2-3 Operation mod (SCSimplicialComplex, Integer)

5.2-4 Operation ^ (SCSimplicialComplex, Integer)

5.2-5 Operation + (SCSimplicialComplex, SCSimplicialComplex)

5.2-6 Operation - (SCSimplicialComplex, SCSimplicialComplex)

5.2-7 Operation * (SCSimplicialComplex, SCSimplicialComplex)

5.2-8 Operation = (SCSimplicialComplex, SCSimplicialComplex)

`SCSimplicialComplex`

5.2-1 Operation + (SCSimplicialComplex, Integer)

5.2-2 Operation - (SCSimplicialComplex, Integer)

5.2-3 Operation mod (SCSimplicialComplex, Integer)

5.2-4 Operation ^ (SCSimplicialComplex, Integer)

5.2-5 Operation + (SCSimplicialComplex, SCSimplicialComplex)

5.2-6 Operation - (SCSimplicialComplex, SCSimplicialComplex)

5.2-7 Operation * (SCSimplicialComplex, SCSimplicialComplex)

5.2-8 Operation = (SCSimplicialComplex, SCSimplicialComplex)

5.3

5.3-1 Operation Union (SCSimplicialComplex, SCSimplicialComplex)

5.3-2 Operation Difference (SCSimplicialComplex, SCSimplicialComplex)

5.3-3 Operation Intersection (SCSimplicialComplex, SCSimplicialComplex)

5.3-4 Size (SCSimplicialComplex)

5.3-5 Length (SCSimplicialComplex)

5.3-6 Operation [] (SCSimplicialComplex)

5.3-7 Iterator (SCSimplicialComplex)

`SCSimplicialComplex`

as a subtype of `Set`

5.3-1 Operation Union (SCSimplicialComplex, SCSimplicialComplex)

5.3-2 Operation Difference (SCSimplicialComplex, SCSimplicialComplex)

5.3-3 Operation Intersection (SCSimplicialComplex, SCSimplicialComplex)

5.3-4 Size (SCSimplicialComplex)

5.3-5 Length (SCSimplicialComplex)

5.3-6 Operation [] (SCSimplicialComplex)

5.3-7 Iterator (SCSimplicialComplex)

`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`

`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.

`‣ 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

`‣ 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

`‣ 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

`‣ 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 ] 48 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 ] 44

`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.

`‣ 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 ] ]

`‣ 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 ] ]

`‣ 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 ] ]

`‣ 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]

`‣ 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" ], [ 113, "RP^3=L(2,1) (VT)" ], [ 589, "(S^2~S^1)#RP^3" ], [ 590, "(S^2xS^1)#RP^3" ], [ 632, "(S^2~S^1)#2#RP^3" ], [ 633, "(S^2xS^1)#2#RP^3" ], [ 2414, "RP^3#RP^3" ], [ 2426, "RP^3=L(2,1) (VT)" ], [ 2488, "(S^2~S^1)#3#RP^3" ], [ 2489, "(S^2xS^1)#3#RP^3" ], [ 2502, "RP^3=L(2,1) (VT)" ], [ 7473, "(S^2~S^1)#4#RP^3" ], [ 7474, "(S^2xS^1)#4#RP^3" ], [ 7504, "(S^2~S^1)#5#RP^3" ], [ 7505, "(S^2xS^1)#5#RP^3" ] ] gap> c:=SCLib.Load(last[1][1]);; gap> SCLib.SearchByName("S^2~S^1"){[1..3]}; [ [ 12, "S^2~S^1 (VT)" ], [ 27, "S^2~S^1 (VT)" ], [ 28, "S^2~S^1 (VT)" ] ] gap> d:=SCLib.Load(last[1][1]);; 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]

`‣ Operation - (SCSimplicialComplex, SCSimplicialComplex)` ( complex1, complex2 ) | ( method ) |

Returns: simplicial complex of type `SCSimplicialComplex`

upon success, `fail`

otherwise.

Calls `SCDifference`

(6.10-5)(`complex1`, `complex2`)

`‣ 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)" ], [ 635, "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]

`‣ 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

`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.

`‣ 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]

`‣ 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

`‣ 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 ] ]

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

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

`‣ 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:=SCLib.Load(last[1][1]);; 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]; [ ]

`‣ 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

`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.

`SCNormalSurface`

As with the object type `SCSimplicialComplex`

, **simpcomp** overloads some standard operations for the object type `SCNormalSurface`

. See a list of overloaded operators below.

`‣ 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 ] ] ]

`‣ 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 ] ] ]

`‣ 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 ] ] ]

`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.

`‣ 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)" ] ] gap> c:=SCLib.Load(last[1][1]);; gap> sl1:=SCNSSlicing(c,[[1,3,5,7,9],[2,4,6,8,10]]);; gap> sl2:=sl1+10;; gap> SCTopologicalType(sl1); "T^2" gap> sl3:=Union(sl1,sl2);; gap> SCTopologicalType(sl3); "T^2 U T^2"

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