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:

Public Member Functions
	GraphTriple (const SFSpace &end0, const SFSpace &centre, const 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 (SFSpace &&end0, SFSpace &&centre, SFSpace &&end1, const Matrix2 &matchingReln0, const Matrix2 &matchingReln1)
	Creates a new graph manifold from three bounded Seifert fibred spaces, which are moved instead of copied. More...

	GraphTriple (const GraphTriple &)=default
	Creates a clone of the given graph manifold. More...

	GraphTriple (GraphTriple &&) noexcept=default
	Moves the contents of the given graph manifold into this new graph manifold. More...

const SFSpace &	end (unsigned which) const
	Returns a reference to one of the two end spaces. More...

const SFSpace &	centre () const
	Returns a reference to the central space. More...

const Matrix2 &	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. 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...

GraphTriple &	operator= (const GraphTriple &)=default
	Sets this to be a clone of the given graph manifold. More...

GraphTriple &	operator= (GraphTriple &&) noexcept=default
	Moves the contents of the given graph manifold into this graph manifold. More...

void	swap (GraphTriple &other) noexcept
	Swaps the contents of this and the given graph manifold. More...

bool	operator== (const GraphTriple &compare) const
	Determines whether this and the given object contain precisely the same presentations of the same graph manifold. More...

bool	operator!= (const GraphTriple &compare) const
	Determines whether this and the given object do not contain precisely the same presentations of the same graph manifold. More...

AbelianGroup	homology () 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...

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.

This class implements C++ move semantics and adheres to the C++ Swappable requirement. It is designed to avoid deep copies wherever possible, even when passing or returning objects by value. Note, however, that GraphTriple still requires a non-trivial (but constant sized) amount of data to be copied even in a move operation.

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.

Constructor & Destructor Documentation

◆ GraphTriple() [1/4]

regina::GraphTriple::GraphTriple	(	const SFSpace &	end0,
		const SFSpace &	centre,
		const SFSpace &	end1,
		const Matrix2 &	matchingReln0,
		const Matrix2 &	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.

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

Exceptions

InvalidArgument One of the spaces end0 and end1 does not have precisely one torus boundary corresponding to a single untwisted puncture in its base orbifold, and/or the space centre does not have precisely two disjoint torus boundaries corresponding to two untwisted punctures in its base orbifold.

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.

◆ GraphTriple() [2/4]

regina::GraphTriple::GraphTriple	(	SFSpace &&	end0,
		SFSpace &&	centre,
		SFSpace &&	end1,
		const Matrix2 &	matchingReln0,
		const Matrix2 &	matchingReln1
	)

inline

Creates a new graph manifold from three bounded Seifert fibred spaces, which are moved instead of copied.

Other than its use of move semantics, this behaves identically to the other constructor that takes the Seifert fibred spaces by const reference.

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

Exceptions

InvalidArgument One of the spaces end0 and end1 does not have precisely one torus boundary corresponding to a single untwisted puncture in its base orbifold, and/or the space centre does not have precisely two disjoint torus boundaries corresponding to two untwisted punctures in its base orbifold.

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.

◆ GraphTriple() [3/4]

regina::GraphTriple::GraphTriple ( const GraphTriple & )

default

Creates a clone of the given graph manifold.

◆ GraphTriple() [4/4]

regina::GraphTriple::GraphTriple ( GraphTriple && )

defaultnoexcept

Moves the contents of the given graph manifold into this new graph manifold.

This is a constant time operation.

The graph manifold that was passed will no longer be usable.

Member Function Documentation

◆ centre()

const SFSpace & regina::GraphTriple::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.

◆ construct()

virtual Triangulation< 3 > regina::Manifold::construct ( ) const

virtualinherited

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

For details of which types of 3-manifolds have implemented this routine, see the class notes for each corresponding subclasses of Manifold.

The default implemention of this routine just throws a NotImplemented exception.

Exceptions

NotImplemented	Explicit construction has not yet been implemented for this particular 3-manifold.
FileError	The construction needs to be read from file (as opposed to computed on the fly), but the file is inaccessible or its contents cannot be read and parsed correctly. Currently this can only happen for the subclass SnapPeaCensusManifold, which reads its triangulations from the SnapPea census databases that are installed with Regina.

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

Reimplemented in regina::Handlebody, regina::LensSpace, regina::SFSpace, regina::SimpleSurfaceBundle, and regina::SnapPeaCensusManifold.

◆ 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.

◆ end()

const SFSpace & regina::GraphTriple::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.

◆ homology()

AbelianGroup regina::GraphTriple::homology ( ) const

overridevirtual

Returns the first homology group of this 3-manifold, if such a routine has been implemented.

For details of which types of 3-manifolds have implemented this routine, see the class notes for each corresponding subclasses of Manifold.

The default implemention of this routine just throws a NotImplemented exception.

Exceptions

NotImplemented	Homology calculation has not yet been implemented for this particular 3-manifold.
FileError	The homology needs to be read from file (as opposed to computed), but the file is inaccessible or its contents cannot be read and parsed correctly. Currently this can only happen for the subclass SnapPeaCensusManifold, which reads its results from the SnapPea census databases that are installed with Regina.

Returns: the first homology group of this 3-manifold, if this functionality has been implemented.

Reimplemented from regina::Manifold.

◆ isHyperbolic()

bool regina::GraphTriple::isHyperbolic ( ) const

inlineoverridevirtual

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::Manifold.

◆ matchingReln()

const Matrix2 & regina::GraphTriple::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.

◆ name()

std::string regina::Manifold::name ( ) const

inherited

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

Returns: the common name of this 3-manifold.

◆ operator!=()

bool regina::GraphTriple::operator!= ( const GraphTriple & compare ) const

inline

Determines whether this and the given object do not contain precisely the same presentations of the same graph manifold.

This routine does not test for homeomorphism. Instead it compares the exact presentations, including the matching matrices and the specific presentations of the bounded Seifert fibred spaces, and determines whether or not these presentations are identical. If you have two different presentations of the same graph manifold, they will be treated as not equal by this routine.

Parameters

compare the presentation with which this will be compared.

Returns: true if and only if this and the given object do not contain identical presentations of the same graph manifold.

◆ operator<() [1/2]

bool regina::GraphTriple::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.

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.

◆ operator<() [2/2]

bool regina::Manifold::operator< ( const Manifold & 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 name().

Parameters

compare the 3-manifold representation with which this will be compared.

Returns: true if and only if this is "smaller" than the given 3-manifold representation.

◆ operator=() [1/2]

GraphTriple & regina::GraphTriple::operator= ( const GraphTriple & )

default

Sets this to be a clone of the given graph manifold.

Returns: a reference to this graph manifold.

◆ operator=() [2/2]

GraphTriple & regina::GraphTriple::operator= ( GraphTriple && )

defaultnoexcept

Moves the contents of the given graph manifold into this graph manifold.

This is a constant time operation.

The graph manifold that was passed will no longer be usable.

Returns: a reference to this graph manifold.

◆ operator==()

bool regina::GraphTriple::operator== ( const GraphTriple & compare ) const

inline

Determines whether this and the given object contain precisely the same presentations of the same graph manifold.

This routine does not test for homeomorphism. Instead it compares the exact presentations, including the matching matrices and the specific presentations of the bounded Seifert fibred spaces, and determines whether or not these presentations are identical. If you have two different presentations of the same graph manifold, they will be treated as not equal by this routine.

Parameters

compare the presentation with which this will be compared.

Returns: true if and only if this and the given object contain identical presentations of the same graph manifold.

◆ 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 use plain ASCII characters where possible, and should not contain any newlines.

Within these limits, this short text ouptut should be as information-rich as possible, since in most cases this forms the basis for the Python __str__() and __repr__() functions.

Python: The Python "stringification" function __str__() will use precisely this function, and for most classes the Python __repr__() function will incorporate this into its output.

Returns: a short text representation of this object.

◆ structure()

std::string regina::Manifold::structure ( ) 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 S³/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.

◆ swap()

void regina::GraphTriple::swap ( GraphTriple & other )

inlinenoexcept

Swaps the contents of this and the given graph manifold.

Parameters

other the graph manifold whose contents should be swapped with this.

◆ texName()

std::string regina::Manifold::texName ( ) 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.

◆ 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 not contain any newlines, and (within these constraints) should be as information-rich as is reasonable.

Unlike str(), this function may use unicode characters to make the output more pleasant to read. The string that is returned will be encoded in UTF-8.

Returns: a short text representation of this object.

◆ writeName()

std::ostream & regina::GraphTriple::writeName ( std::ostream & out ) const

overridevirtual

Writes the common name of this 3-manifold as a human-readable string to the given output stream.

Python: Not present. Instead use the variant name() that takes no arguments and returns a string.

Parameters

out	the output stream to which to write.

Returns: a reference to the given output stream.

Implements regina::Manifold.

◆ writeStructure()

std::ostream & regina::Manifold::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 S³/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: Not present. Instead use the variant structure() that takes no arguments and returns a string.

Parameters

out	the output stream to which to write.

Returns: a reference to the given output stream.

Reimplemented in regina::SFSpace, and regina::SnapPeaCensusManifold.

◆ writeTeXName()

std::ostream & regina::GraphTriple::writeTeXName ( std::ostream & out ) const

overridevirtual

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: Not present. Instead use the variant texName() that takes no arguments and returns a string.

Parameters

out	the output stream to which to write.

Returns: a reference to the given output stream.

Implements regina::Manifold.

◆ writeTextLong()

void regina::Manifold::writeTextLong ( std::ostream & out ) const

inlineinherited

Writes a detailed text representation of this object to the given output stream.

Subclasses must not override this routine. They should override writeName() and writeStructure() instead.

Python: Not present. Use detail() instead.

Parameters

out	the output stream to which to write.

◆ writeTextShort()

void regina::Manifold::writeTextShort ( std::ostream & out ) const

inlineinherited

Writes a short text representation of this object to the given output stream.

Subclasses must not override this routine. They should override writeName() instead.

Python: Not present. Use str() instead.

Parameters

out	the output stream to which to write.

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

manifold/graphtriple.h

Public Member Functions

Detailed Description

Constructor & Destructor Documentation

◆ GraphTriple() [1/4]

◆ GraphTriple() [2/4]

◆ GraphTriple() [3/4]

◆ GraphTriple() [4/4]

Member Function Documentation

◆ centre()

◆ construct()

◆ detail()

◆ end()

◆ homology()

◆ isHyperbolic()

◆ matchingReln()

◆ name()

◆ operator!=()

◆ operator<() [1/2]

◆ operator<() [2/2]

◆ operator=() [1/2]

◆ operator=() [2/2]

◆ operator==()

◆ str()

◆ structure()

◆ swap()

◆ texName()

◆ utf8()

◆ writeName()

◆ writeStructure()

◆ writeTeXName()

◆ writeTextLong()

◆ writeTextShort()