Regina Calculation Engine
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regina::GraphTriple Class Reference

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

#include <manifold/graphtriple.h>

Inheritance diagram for regina::GraphTriple:
regina::Manifold regina::Output< Manifold >

Public Member Functions

 GraphTriple (SFSpace *end0, SFSpace *centre, SFSpace *end1, const Matrix2 &matchingReln0, const Matrix2 &matchingReln1)
 Creates a new graph manifold from three bounded Seifert fibred spaces, as described in the class notes. More...
 
 GraphTriple (const GraphTriple &cloneMe)
 Creates a clone of the given graph manifold. More...
 
 ~GraphTriple ()
 Destroys this structure along with the component Seifert fibred spaces and matching matrices. More...
 
const SFSpaceend (unsigned which) const
 Returns a reference to one of the two end spaces. More...
 
const SFSpacecentre () const
 Returns a reference to the central space. More...
 
const Matrix2matchingReln (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 GraphTriple &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...
 
GraphTripleoperator= (const GraphTriple &cloneMe)
 Sets this to be a clone of the given graph manifold. More...
 
AbelianGrouphomology () const override
 Returns the first homology group of this 3-manifold, if such a routine has been implemented. More...
 
bool isHyperbolic () const override
 Returns whether or not this is a finite-volume hyperbolic manifold. More...
 
std::ostream & writeName (std::ostream &out) const override
 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 override
 Writes the common name of this 3-manifold in TeX format to the given output stream. More...
 
std::string name () const
 Returns the common name of this 3-manifold as a human-readable string. More...
 
std::string TeXName () const
 Returns the common name of this 3-manifold in TeX format. More...
 
std::string structure () const
 Returns details of the structure of this 3-manifold that might not be evident from its common name. More...
 
virtual Triangulation< 3 > * construct () const
 Returns a triangulation of this 3-manifold, if such a construction has been implemented. More...
 
AbelianGrouphomologyH1 () const
 Returns the first homology group of this 3-manifold, if such a routine has been implemented. More...
 
bool operator< (const Manifold &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...
 
void writeTextShort (std::ostream &out) const
 Writes a short text representation of this object to the given output stream. More...
 
void writeTextLong (std::ostream &out) const
 Writes a detailed text representation of this object to the given output stream. More...
 
std::string str () const
 Returns a short text representation of this object. More...
 
std::string utf8 () const
 Returns a short text representation of this object using unicode characters. More...
 
std::string detail () const
 Returns a detailed text representation of this object. 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 Manifold routine homology() 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 name() and TeXName() 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.

Member Function Documentation

◆ detail()

std::string regina::Output< Manifold , false >::detail ( ) const
inherited

Returns a detailed text representation of this object.

This text may span many lines, and should provide the user with all the information they could want. It should be human-readable, should not contain extremely long lines (which cause problems for users reading the output in a terminal), and should end with a final newline. There are no restrictions on the underlying character set.

Returns
a detailed text representation of this object.

◆ str()

std::string regina::Output< Manifold , false >::str ( ) const
inherited

Returns a short text representation of this object.

This text should be human-readable, should fit on a single line, and should not end with a newline. Where possible, it should use plain ASCII characters.

Python
In addition to str(), this is also used as the Python "stringification" function str().
Returns
a short text representation of this object.

◆ utf8()

std::string regina::Output< Manifold , false >::utf8 ( ) const
inherited

Returns a short text representation of this object using unicode characters.

Like str(), this text should be human-readable, should fit on a single line, and should not end with a newline. In addition, it may use unicode characters to make the output more pleasant to read. This string will be encoded in UTF-8.

Returns
a short text representation of this object.

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

Copyright © 1999-2021, 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@maths.uq.edu.au).