regina::NGraphTriple Class Reference
[Standard 3-Manifolds]

Represents a closed graph manifold formed by joining three bounded Seifert fibred spaces along their torus boundaries. More...

#include <ngraphtriple.h>

Inheritance diagram for regina::NGraphTriple:

regina::NManifold regina::ShareableObject regina::boost::noncopyable List of all members.

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 NSFSpaceend (unsigned which) const
 Returns a reference to one of the two end spaces.
const NSFSpacecentre () const
 Returns a reference to the central space.
const NMatrix2matchingReln (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.
NAbelianGroupgetHomologyH1 () 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.

Detailed Description

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.

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.

Todo:
Optimise: Speed up homology calculations involving orientable base spaces by adding rank afterwards, instead of adding generators for genus into the presentation matrix.


Constructor & Destructor Documentation

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.

Precondition:
Spaces end0 and end1 each have a single torus boundary, corresponding to a single untwisted puncture in the base orbifold.

Space centre has two disjoint torus boundaries, corresponding to two untwisted punctures in the base orbifold.

Each of the given matrices has determinant +1 or -1.

Parameters:
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.


Member Function Documentation

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.

Returns:
a reference to the requested Seifert fibred space.

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.

Parameters:
which 0 if the first end space is to be returned, or 1 if the second end space is to be returned.
Returns:
a reference to the requested Seifert fibred space.

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.

Returns:
the first homology group of this 3-manifold, or 0 if the appropriate calculation routine has not yet been implemented.

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).

Parameters:
which indicates which particular join should be examined; this should be 0 or 1 as described above.
Returns:
a reference to the requested matching matrix.

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.

Parameters:
compare the representation with which this will be compared.
Returns:
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.

Python:
The parameter out does not exist; standard output will be used.
Parameters:
out the output stream to which to write.
Returns:
a reference to the given output stream.

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.

Warning:
The behaviour of this routine has changed as of Regina 4.3; in earlier versions, leading and trailing dollar signs were provided.
Python:
The parameter out does not exist; standard output will be used.
Parameters:
out the output stream to which to write.
Returns:
a reference to the given output stream.

Implements regina::NManifold.


The documentation for this class was generated from the following file:
Copyright © 1999-2006, Ben Burton
This software is released under the GNU General Public License.
For further information, or to submit a bug or other problem, please contact Ben Burton (bab@debian.org).