Regina Calculation Engine
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regina::NGraphTriple Class Reference

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

#include <manifold/ngraphtriple.h>

Inheritance diagram for regina::NGraphTriple:
regina::NManifold regina::ShareableObject regina::boost::noncopyable

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. More...
 
 ~NGraphTriple ()
 Destroys this structure along with the component Seifert fibred spaces and matching matrices. More...
 
const NSFSpaceend (unsigned which) const
 Returns a reference to one of the two end spaces. More...
 
const NSFSpacecentre () const
 Returns a reference to the central space. More...
 
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. More...
 
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. More...
 
NAbelianGroupgetHomologyH1 () const
 Returns the first homology group of this 3-manifold, if such a routine has been implemented. More...
 
bool isHyperbolic () const
 Returns whether or not this is a finite-volume hyperbolic manifold. More...
 
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. More...
 
std::ostream & writeTeXName (std::ostream &out) const
 Writes the common name of this 3-manifold in TeX format to the given output stream. More...
 
std::string getName () const
 Returns the common name of this 3-manifold as a human-readable string. More...
 
std::string getTeXName () const
 Returns the common name of this 3-manifold in TeX format. More...
 
std::string getStructure () const
 Returns details of the structure of this 3-manifold that might not be evident from its common name. More...
 
virtual NTriangulationconstruct () const
 Returns a triangulation of this 3-manifold, if such a construction has been implemented. More...
 
bool operator< (const NManifold &compare) const
 Determines in a fairly ad-hoc fashion whether this representation of this 3-manifold is "smaller" than the given representation of the given 3-manifold. More...
 
virtual std::ostream & writeStructure (std::ostream &out) const
 Writes details of the structure of this 3-manifold that might not be evident from its common name to the given output stream. More...
 
virtual void writeTextShort (std::ostream &out) const
 Writes this object in short text format to the given output stream. More...
 
virtual void writeTextLong (std::ostream &out) const
 Writes this object in long text format to the given output stream. More...
 
Input and Output
std::string str () const
 Returns the output from writeTextShort() as a string. More...
 
std::string toString () const
 A deprecated alias for str(), which returns the output from writeTextShort() as a string. More...
 
std::string detail () const
 Returns the output from writeTextLong() as a string. More...
 
std::string toStringLong () const
 A deprecated alias for detail(), which returns the output from writeTextLong() as a string. More...
 

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. 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.

Warning
The 2-by-2 matrices used in this class are not the same matrices that appear in the manifold name returned by getName() and getTeXName() and seen in the census data files. The matrices used in this class work from the inside out, describing the boundary torus on each end space in terms of a boundary torus on the central space. The matrices used in the manifold name work from left to right in the diagram above, describing a boundary torus on the central space or rightmost end space in terms of a boundary torus on the leftmost end space or central space respectively. The upshot of all this is that the first matrix becomes inverted (and the second matrix remains unchanged). It is likely that future versions of Regina will replace this class with a more general class that (amongst other things) removes this inconsistency.
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
end0the first end space, as described in the class notes.
centrethe central space, as described in the class notes.
end1the second end space, as described in the class notes.
matchingReln0the 2-by-2 matching matrix that specifies how spaces end0 and centre are joined.
matchingReln1the 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.
NTriangulation * regina::NManifold::construct ( ) const
inlinevirtualinherited

Returns a triangulation of this 3-manifold, if such a construction has been implemented.

If no construction routine has yet been implemented for this 3-manifold (for instance, if this 3-manifold is a Seifert fibred space with sufficiently many exceptional fibres) then this routine will return 0.

The details of which 3-manifolds have construction routines can be found in the notes for the corresponding subclasses of NManifold. The default implemention of this routine returns 0.

Returns
a triangulation of this 3-manifold, or 0 if the appropriate construction routine has not yet been implemented.

Reimplemented in regina::NSFSpace, regina::NSnapPeaCensusManifold, regina::NLensSpace, and regina::NSimpleSurfaceBundle.

std::string regina::ShareableObject::detail ( ) const
inherited

Returns the output from writeTextLong() as a string.

Returns
a long text representation of this object.
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
which0 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.

std::string regina::NManifold::getName ( ) const
inherited

Returns the common name of this 3-manifold as a human-readable string.

Returns
the common name of this 3-manifold.
std::string regina::NManifold::getStructure ( ) const
inherited

Returns details of the structure of this 3-manifold that might not be evident from its common name.

For instance, for an orbit space S^3/G this routine might return the full Seifert structure.

This routine may return the empty string if no additional details are deemed necessary.

Returns
a string describing additional structural details.
std::string regina::NManifold::getTeXName ( ) const
inherited

Returns the common name of this 3-manifold in TeX format.

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.
Returns
the common name of this 3-manifold in TeX format.
bool regina::NGraphTriple::isHyperbolic ( ) const
inlinevirtual

Returns whether or not this is a finite-volume hyperbolic manifold.

Returns
true if this is a finite-volume hyperbolic manifold, or false if not.

Implements 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
whichindicates 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::NManifold::operator< ( const NManifold compare) const
inherited

Determines in a fairly ad-hoc fashion whether this representation of this 3-manifold is "smaller" than the given representation of the given 3-manifold.

The ordering imposed on 3-manifolds is purely aesthetic on the part of the author, and is subject to change in future versions of Regina.

The ordering also depends on the particular representation of the 3-manifold that is used. As an example, different representations of the same Seifert fibred space might well be ordered differently.

All that this routine really offers is a well-defined way of ordering 3-manifold representations.

Warning
Currently this routine is only implemented in full for closed 3-manifolds. For most classes of bounded 3-manifolds, this routine simply compares the strings returned by getName().
Parameters
comparethe 3-manifold representation with which this will be compared.
Returns
true if and only if this is "smaller" than the given 3-manifold representation.
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
comparethe representation with which this will be compared.
Returns
true if and only if this is "smaller" than the given graph manifold representation.
std::string regina::ShareableObject::str ( ) const
inherited

Returns the output from writeTextShort() as a string.

Python:
This implements the __str__() function.
Returns
a short text representation of this object.
std::string regina::ShareableObject::toString ( ) const
inlineinherited

A deprecated alias for str(), which returns the output from writeTextShort() as a string.

Deprecated:
This routine has (at long last) been deprecated; use the simpler-to-type str() instead.
Returns
a short text representation of this object.
std::string regina::ShareableObject::toStringLong ( ) const
inlineinherited

A deprecated alias for detail(), which returns the output from writeTextLong() as a string.

Deprecated:
This routine has (at long last) been deprecated; use the simpler-to-type detail() instead.
Returns
a long text representation of this object.
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
outthe output stream to which to write.
Returns
a reference to the given output stream.

Implements regina::NManifold.

std::ostream & regina::NManifold::writeStructure ( std::ostream &  out) const
inlinevirtualinherited

Writes details of the structure of this 3-manifold that might not be evident from its common name to the given output stream.

For instance, for an orbit space S^3/G this routine might write the full Seifert structure.

This routine may write nothing if no additional details are deemed necessary. The default implementation of this routine behaves in this way.

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

Reimplemented in regina::NSFSpace, and regina::NSnapPeaCensusManifold.

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
outthe output stream to which to write.
Returns
a reference to the given output stream.

Implements regina::NManifold.

void regina::NManifold::writeTextLong ( std::ostream &  out) const
inlinevirtualinherited

Writes this object in long text format to the given output stream.

The output should provide the user with all the information they could want. The output should be human-readable, should not contain extremely long lines (so users can read the output in a terminal), and should end with a final newline.

The default implementation of this routine merely calls writeTextShort() and adds a newline.

Python:
The parameter out does not exist; standard output will be used.
Parameters
outthe output stream to which to write.

Reimplemented from regina::ShareableObject.

void regina::NManifold::writeTextShort ( std::ostream &  out) const
inlinevirtualinherited

Writes this object in short text format to the given output stream.

The output should be human-readable, should fit on a single line, and should not end with a newline.

Python:
The parameter out does not exist; standard output will be used.
Parameters
outthe output stream to which to write.

Implements regina::ShareableObject.


The documentation for this class was generated from the following file:

Copyright © 1999-2014, The Regina development team
This software is released under the GNU General Public License, with some additional permissions; see the source code for details.
For further information, or to submit a bug or other problem, please contact Ben Burton (bab@debian.org).