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

Represents a closed graph manifold formed by joining two bounded Seifert fibred spaces along a common torus. More...

#include <ngraphpair.h>

Inheritance diagram for regina::NGraphPair:

regina::NManifold regina::ShareableObject regina::boost::noncopyable

List of all members.

Public Member Functions

 NGraphPair (NSFSpace *sfs0, NSFSpace *sfs1, long mat00, long mat01, long mat10, long mat11)
 Creates a new graph manifold as a pair of joined Seifert fibred spaces.
 NGraphPair (NSFSpace *sfs0, NSFSpace *sfs1, const NMatrix2 &matchingReln)
 Creates a new graph manifold as a pair of joined Seifert fibred spaces.
 ~NGraphPair ()
 Destroys this structure along with the component Seifert fibred spaces and the matching matrix.
const NSFSpacesfs (unsigned which) const
 Returns a reference to one of the two bounded Seifert fibred spaces that are joined together.
const NMatrix2matchingReln () const
 Returns a reference to the 2-by-2 matrix describing how the two Seifert fibred spaces are joined together.
bool operator< (const NGraphPair &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 two bounded Seifert fibred spaces along a common torus.

Each Seifert fibred space must have just one boundary component, corresponding to a puncture in the base orbifold (with no fibre-reversing twist as one travels around this boundary).

The way in which the two spaces are joined is specified by a 2-by-2 matrix M. This matrix expresses the locations of the fibres and base orbifold of the second Seifert fibred space in terms of the first.

More specifically, suppose that f0 and o0 are generators of the common torus, where f0 represents a directed fibre in the first Seifert fibred space and o0 represents the oriented boundary of the corresponding base orbifold. Likewise, let f1 and o1 be generators of the common torus representing a directed fibre and the base orbifold of the second Seifert fibred space. Then the curves f0, o0, f1 and o1 are related as follows:

     [f1]       [f0]
     [  ] = M * [  ]
     [o1]       [o0]
 

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.

Test:
Tested in the test suite, though not exhaustively.
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::NGraphPair::NGraphPair ( NSFSpace sfs0,
NSFSpace sfs1,
long  mat00,
long  mat01,
long  mat10,
long  mat11 
) [inline]

Creates a new graph manifold as a pair of joined Seifert fibred spaces.

The two bounded Seifert fibred spaces and the four elements of the 2-by-2 matching matrix are all passed separately. The elements of the matching matrix combine to give the full matrix M as follows:

           [ mat00  mat01 ]
     M  =  [              ]
           [ mat10  mat11 ]
 

Note that the new object will take ownership of the two given Seifert fibred spaces, and when this object is destroyed the Seifert fibred spaces will be destroyed also.

Precondition:
Each Seifert fibred space has a single torus boundary, corresponding to a single untwisted puncture in the base orbifold.

The given matching matrix has determinant +1 or -1.

Parameters:
sfs0 the first Seifert fibred space.
sfs1 the second Seifert fibred space.
mat00 the (0,0) element of the matching matrix.
mat01 the (0,1) element of the matching matrix.
mat10 the (1,0) element of the matching matrix.
mat11 the (1,1) element of the matching matrix.

regina::NGraphPair::NGraphPair ( NSFSpace sfs0,
NSFSpace sfs1,
const NMatrix2 matchingReln 
) [inline]

Creates a new graph manifold as a pair of joined Seifert fibred spaces.

The two bounded Seifert fibred spaces and the entire 2-by-2 matching matrix are each passed separately.

Note that the new object will take ownership of the two given Seifert fibred spaces, and when this object is destroyed the Seifert fibred spaces will be destroyed also.

Precondition:
Each Seifert fibred space has a single torus boundary, corresponding to a single untwisted puncture in the base orbifold.

The given matching matrix has determinant +1 or -1.

Parameters:
sfs0 the first Seifert fibred space.
sfs1 the second Seifert fibred space.
matchingReln the 2-by-2 matching matrix.

regina::NGraphPair::~NGraphPair (  ) 

Destroys this structure along with the component Seifert fibred spaces and the matching matrix.


Member Function Documentation

NAbelianGroup* regina::NGraphPair::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::NGraphPair::matchingReln (  )  const [inline]

Returns a reference to the 2-by-2 matrix describing how the two Seifert fibred spaces are joined together.

See the class notes for details on precisely how this matrix is represented.

Returns:
a reference to the matching matrix.

bool regina::NGraphPair::operator< ( const NGraphPair 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.

const NSFSpace & regina::NGraphPair::sfs ( unsigned  which  )  const [inline]

Returns a reference to one of the two bounded Seifert fibred spaces that are joined together.

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

std::ostream& regina::NGraphPair::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::NGraphPair::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 © 1999-2009, 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).