Represents a closed graph manifold formed by joining three bounded Seifert fibred spaces along their torus boundaries. More...
#include <ngraphtriple.h>
Public Member Functions | |
NGraphTriple (NSFSpace *end0, NSFSpace *centre, NSFSpace *end1, const NMatrix2 &matchingReln0, const NMatrix2 &matchingReln1) | |
Creates a new graph manifold from three bounded Seifert fibred spaces, as described in the class notes. | |
~NGraphTriple () | |
Destroys this structure along with the component Seifert fibred spaces and matching matrices. | |
const NSFSpace & | end (unsigned which) const |
Returns a reference to one of the two end spaces. | |
const NSFSpace & | centre () const |
Returns a reference to the central space. | |
const NMatrix2 & | matchingReln (unsigned which) const |
Returns a reference to the 2-by-2 matrix describing how the two requested bounded Seifert fibred spaces are joined together. | |
bool | operator< (const NGraphTriple &compare) const |
Determines in a fairly ad-hoc fashion whether this representation of this space is "smaller" than the given representation of the given space. | |
NAbelianGroup * | getHomologyH1 () const |
Returns the first homology group of this 3-manifold, if such a routine has been implemented. | |
std::ostream & | writeName (std::ostream &out) const |
Writes the common name of this 3-manifold as a human-readable string to the given output stream. | |
std::ostream & | writeTeXName (std::ostream &out) const |
Writes the common name of this 3-manifold in TeX format to the given output stream. |
Represents a closed graph manifold formed by joining three bounded Seifert fibred spaces along their torus boundaries.
There must be one Seifert fibred space at either end, each with a single torus boundary (corresponding to a single puncture in the base orbifold, with no fibre-reversing twist around this puncture). Each of these end spaces is joined to the space in the centre, which has two disjoint torus boundaries (corresponding to two punctures in the base orbifold, again with no fibre-reversing twists around these punctures).
This configuration is illustrated in the diagram below. The large boxes represent the bounded Seifert fibred spaces, and the small tunnels show how their boundaries are joined.
/---------------\ /-----------------\ /---------------\ | | | | | | | End space 0 --- Central space --- End space 1 | | --- --- | | | | | | | \---------------/ \-----------------/ \---------------/
The way in which each pair of spaces is joined is specified by a 2-by-2 matrix. This matrix expresses the locations of the fibres and base orbifold of the corresponding end space in terms of the central space. Note that these are not the same matrices that appear in the manifold name in the census data files! See the warning below.
More specifically, consider the matrix M that describes the joining of the central space and the first end space (marked above as end space 0). Suppose that f and o are generators of the common boundary torus, where f represents a directed fibre in the central space and o represents the oriented boundary of the corresponding base orbifold. Likewise, let f0 and o0 be generators of the common boundary torus representing a directed fibre and the base orbifold of the first end space. Then the curves f, o, f0 and o0 are related as follows:
[f0] [f ] [ ] = M * [ ] [o0] [o ]
Likewise, let matrix M' describe the joining of the central space and the second end space (marked in the diagram above as end space 1). Let f' and o' be curves on the common boundary torus representing the fibres and the base orbifold of the central space, and let f1 and o1 be curves on this same torus representing the fibres and the base orbifold of the second end space. Then the curves f', o', f1 and o1 are related as follows:
[f1] [f'] [ ] = M' * [ ] [o1] [o']
See the page on Notation for Seifert fibred spaces for details on some of the terminology used above.
The optional NManifold routine getHomologyH1() is implemented, but the optional routine construct() is not.
regina::NGraphTriple::NGraphTriple | ( | NSFSpace * | end0, |
NSFSpace * | centre, | ||
NSFSpace * | end1, | ||
const NMatrix2 & | matchingReln0, | ||
const NMatrix2 & | matchingReln1 | ||
) | [inline] |
Creates a new graph manifold from three bounded Seifert fibred spaces, as described in the class notes.
The three Seifert fibred spaces and both 2-by-2 matching matrices are passed separately.
Note that the new object will take ownership of the three given Seifert fibred spaces, and when this object is destroyed the Seifert fibred spaces will be destroyed also.
end0 | the first end space, as described in the class notes. |
centre | the central space, as described in the class notes. |
end1 | the second end space, as described in the class notes. |
matchingReln0 | the 2-by-2 matching matrix that specifies how spaces end0 and centre are joined. |
matchingReln1 | the 2-by-2 matching matrix that specifies how spaces end1 and centre are joined. |
regina::NGraphTriple::~NGraphTriple | ( | ) |
Destroys this structure along with the component Seifert fibred spaces and matching matrices.
const NSFSpace & regina::NGraphTriple::centre | ( | ) | const [inline] |
Returns a reference to the central space.
This is the Seifert fibred space with two boundary components, to which the two end spaces are joined. See the class notes for further discussion.
const NSFSpace & regina::NGraphTriple::end | ( | unsigned | which ) | const [inline] |
Returns a reference to one of the two end spaces.
These are the Seifert fibred spaces with just one boundary component, to be joined to the central space. See the class notes for further discussion.
which | 0 if the first end space is to be returned, or 1 if the second end space is to be returned. |
NAbelianGroup* regina::NGraphTriple::getHomologyH1 | ( | ) | const [virtual] |
Returns the first homology group of this 3-manifold, if such a routine has been implemented.
If the calculation of homology has not yet been implemented for this 3-manifold then this routine will return 0.
The details of which 3-manifolds have homology calculation routines can be found in the notes for the corresponding subclasses of NManifold. The default implemention of this routine returns 0.
The homology group will be newly allocated and must be destroyed by the caller of this routine.
Reimplemented from regina::NManifold.
const NMatrix2 & regina::NGraphTriple::matchingReln | ( | unsigned | which ) | const [inline] |
Returns a reference to the 2-by-2 matrix describing how the two requested bounded Seifert fibred spaces are joined together.
See the class notes for details on precisely how these matrices are represented.
The argument which indicates which particular join should be examined. A value of 0 denotes the join between the central space and the first end space (corresponding to matrix M in the class notes), whereas a value of 1 denotes the join between the central space and the second end space (corresponding to matrix M' in the class notes).
which | indicates which particular join should be examined; this should be 0 or 1 as described above. |
bool regina::NGraphTriple::operator< | ( | const NGraphTriple & | compare ) | const |
Determines in a fairly ad-hoc fashion whether this representation of this space is "smaller" than the given representation of the given space.
The ordering imposed on graph manifolds is purely aesthetic on the part of the author, and is subject to change in future versions of Regina. It also depends upon the particular representation, so that different representations of the same space may be ordered differently.
All that this routine really offers is a well-defined way of ordering graph manifold representations.
compare | the representation with which this will be compared. |
true
if and only if this is "smaller" than the given graph manifold representation. std::ostream& regina::NGraphTriple::writeName | ( | std::ostream & | out ) | const [virtual] |
Writes the common name of this 3-manifold as a human-readable string to the given output stream.
out | the output stream to which to write. |
Implements regina::NManifold.
std::ostream& regina::NGraphTriple::writeTeXName | ( | std::ostream & | out ) | const [virtual] |
Writes the common name of this 3-manifold in TeX format to the given output stream.
No leading or trailing dollar signs will be included.
out | the output stream to which to write. |
Implements regina::NManifold.