Represents a blocked sequence of three Seifert fibred spaces joined along connecting tori. More...

#include <subcomplex/blockedsfstriple.h>

Inheritance diagram for regina::BlockedSFSTriple:

Public Member Functions
	BlockedSFSTriple (const BlockedSFSTriple &src)=default
	Creates a new copy of the given structure. More...

	BlockedSFSTriple (BlockedSFSTriple &&src) noexcept=default
	Moves the contents of the given structure into this new structure. More...

BlockedSFSTriple &	operator= (const BlockedSFSTriple &src)=default
	Sets this to be a copy of the given structure. More...

BlockedSFSTriple &	operator= (BlockedSFSTriple &&src) noexcept=default
	Moves the contents of the given structure into this structure. More...

void	swap (BlockedSFSTriple &other) noexcept
	Swaps the contents of this and the given structure. More...

const SatRegion &	end (int which) const
	Returns details of the requested end region, as described in the class notes above. More...

const SatRegion &	centre () const
	Returns details of the central saturated region, as described in the class notes above. More...

const Matrix2 &	matchingReln (int which) const
	Returns the matrix describing how the given end region is joined to the central region. More...

bool	operator== (const BlockedSFSTriple &other) const
	Determines whether this and the given structure represent the same type of blocked sequence of three Seifert fibred spaces. More...

bool	operator!= (const BlockedSFSTriple &other) const
	Determines whether this and the given structure do not represent the same type of blocked sequence of three Seifert fibred spaces. More...

std::unique_ptr< Manifold >	manifold () const override
	Returns the 3-manifold represented by this triangulation, if such a recognition routine has been implemented. More...

std::ostream &	writeName (std::ostream &out) const override
	Writes the name of this triangulation as a human-readable string to the given output stream. More...

std::ostream &	writeTeXName (std::ostream &out) const override
	Writes the name of this triangulation in TeX format to the given output stream. More...

void	writeTextLong (std::ostream &out) const override
	Writes a detailed text representation of this object to the given output stream. More...

std::string	name () const
	Returns the name of this specific triangulation as a human-readable string. More...

std::string	texName () const
	Returns the name of this specific triangulation in TeX format. More...

virtual AbelianGroup	homology () const
	Returns the expected first homology group of this triangulation, if such a routine has been implemented. More...

virtual void	writeTextShort (std::ostream &out) const
	Writes a short 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...

Static Public Member Functions
static std::unique_ptr< BlockedSFSTriple >	recognise (const Triangulation< 3 > &tri)
	Determines if the given triangulation is a blocked sequence of three Seifert fibred spaces, as described in the class notes above. More...

static std::unique_ptr< StandardTriangulation >	recognise (Component< 3 > *component)
	Determines whether the given component represents one of the standard triangulations understood by Regina. More...

Detailed Description

Represents a blocked sequence of three Seifert fibred spaces joined along connecting tori.

This is a particular type of triangulation of a graph manifold, formed from three saturated regions whose various torus boundaries are identified as described below. Optional layerings may be placed between torus boundaries to allow for more interesting relationships between the respective boundary curves of each region. For more detail on saturated regions and their constituent saturated blocks, see the SatRegion class; for more detail on layerings, see the Layering class.

The three saturated regions must be joined together as illustrated below. Each large box represents a saturated region, and the small tunnels show where the region boundaries are joined (possibly via layerings).

    /----------------\   /------------------\   /----------------\
    |                |   |                  |   |                |
    |  End region 0   ---   Central region   ---   End region 1  |
    |                 ---                    ---                 |
    |                |   |                  |   |                |
    ----------------/   ------------------/   ----------------/

Each of the end regions must have precisely one boundary component formed from just one saturated annulus. The central region may have two boundary components formed from one saturated annulus each. Alternatively, it may have one boundary formed from two saturated annuli, where this boundary is pinched together so that each annulus becomes a two-sided torus joined to one of the end regions. None of the boundary components (or the two-sided tori discussed above) may be twisted (i.e., they must be tori, not Klein bottles).

The ways in which the various region boundaries are identified are specified by 2-by-2 matrices, which express curves representing the fibres and base orbifold of each end region in terms of the central region (see the page on Notation for Seifert fibred spaces for terminology).

Specifically, consider the matrix M that describes the joining of the central region and the first end region (marked in the diagram above as end region 0). Suppose that f and o are directed curves on the central region boundary and f0 and o0 are directed curves on the first end region boundary, where f and f0 represent the fibres of each region and o and o0 represent the base orbifolds. Then the boundaries are joined according to the following relation:

    [f0]       [f ]
    [  ] = M * [  ]
    [o0]       [o ]

Likewise, let M' be the matrix describing how the central region and the second end region (marked in the diagram as end region 1) are joined. Let f' and o' be directed curves on the other central region boundary and f1 and o1 be directed curves on the second end region boundary, where f' and f1 represent fibres and o and o1 represent the base orbifolds. Then the boundaries are joined according to the relation:

    [f1]        [f']
    [  ] = M' * [  ]
    [o1]        [o']

If a layering is present between two regions, then the corresponding boundary curves are not identified directly. In this case, the relevant matrix M or M' shows how the layering relates the curves on each region boundary.

Note that the routines writeName() and writeTeXName() do not offer enough information to uniquely identify the triangulation, since this essentially requires 2-dimensional assemblings of saturated blocks. For full details, writeTextLong() may be used instead.

The optional StandardTriangulation routine manifold() is implemented for this class, but homology() 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 the only way to create objects of this class (aside from copying or moving) is via the static member function recognise().

Constructor & Destructor Documentation

◆ BlockedSFSTriple() [1/2]

regina::BlockedSFSTriple::BlockedSFSTriple ( const BlockedSFSTriple & src )

default

Creates a new copy of the given structure.

This will induce a deep copy of src.

Parameters

src	the structure to copy.

◆ BlockedSFSTriple() [2/2]

regina::BlockedSFSTriple::BlockedSFSTriple ( BlockedSFSTriple && src )

defaultnoexcept

Moves the contents of the given structure into this new structure.

This is a constant time operation.

The structure that was passed (src) will no longer be usable.

Parameters

src	the structure to move from.

Member Function Documentation

◆ centre()

const SatRegion & regina::BlockedSFSTriple::centre ( ) const

inline

Returns details of the central saturated region, as described in the class notes above.

This is the saturated region with two boundary annuli, each of which is joined to one of the end regions.

Returns: details of the central region.

◆ detail()

std::string regina::Output< StandardTriangulation , 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 SatRegion & regina::BlockedSFSTriple::end ( int which ) const

inline

Returns details of the requested end region, as described in the class notes above.

The end regions are the two saturated regions with one boundary annulus each, which are both joined to the central region.

Parameters

which 0 if the first end region should be returned (marked as end region 0 in the class notes), or 1 if the second end region should be returned (marked as end region 1 in the class notes).

Returns: details of the requested end region.

◆ homology()

virtual AbelianGroup regina::StandardTriangulation::homology ( ) const

virtualinherited

Returns the expected first homology group of this triangulation, if such a routine has been implemented.

This routine does not work by calling Triangulation<3>::homology() on the associated real triangulation. Instead the homology is calculated directly from the known properties of this standard triangulation.

This means that homology() needs to be implemented separately for each class of standard triangulation. See the class notes for each subclass of StandardTriangulation for details on whether homology has been implemented for that particular subclass. The default implementation of this routine just throws a NotImplemented exception.

Most users will not need this routine, since presumably you already have an explicit Triangulation<3> available and so you can just call Triangulation<3>::homology() instead (which, unlike this routine, is always implemented). This StandardTriangulation::homology() routine should be seen as more of a verification/validation tool for the Regina developers.

If this StandardTriangulation describes an entire Triangulation<3> (and not just a part thereof) then the results of this routine should be identical to the homology group obtained by calling Triangulation<3>::homology() upon the associated real triangulation.

Exceptions

NotImplemented	Homology calculation has not yet been implemented for this particular type of standard triangulation.
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 SnapPeaCensusTri, which reads its results from the SnapPea census databases that are installed with Regina.

Returns: the first homology group of this triangulation, if this functionality has been implemented.

Reimplemented in regina::LayeredChain, regina::LayeredChainPair, regina::LayeredLensSpace, regina::LayeredLoop, regina::LayeredSolidTorus, regina::LayeredTorusBundle, regina::SnapPeaCensusTri, regina::SnappedBall, regina::SpiralSolidTorus, regina::TriSolidTorus, and regina::TrivialTri.

◆ manifold()

std::unique_ptr< Manifold > regina::BlockedSFSTriple::manifold ( ) const

overridevirtual

Returns the 3-manifold represented by this triangulation, if such a recognition routine has been implemented.

If the 3-manifold cannot be recognised then this routine will return null.

The details of which standard triangulations have 3-manifold recognition routines can be found in the notes for the corresponding subclasses of StandardTriangulation. The default implementation of this routine returns null.

It is expected that the number of triangulations whose underlying 3-manifolds can be recognised will grow between releases.

Returns: the underlying 3-manifold.

Reimplemented from regina::StandardTriangulation.

◆ matchingReln()

const Matrix2 & regina::BlockedSFSTriple::matchingReln ( int which ) const

inline

Returns the matrix describing how the given end region is joined to the central region.

Note that if a layering is placed between the two respective region boundaries, then any changes to the boundary relationships caused by the layering are included in this matrix.

See the class notes above for precise information on how each matrix is presented.

Parameters

which 0 if the matrix returned should describe how the central region is joined to the first end region (marked end region 0 in the class notes), or 1 if the matrix returned should describe how the central region is joined to the second end region (marked end region 1 in the class notes).

Returns: the matrix describing how the requested region boundaries are joined.

◆ name()

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

inherited

Returns the name of this specific triangulation as a human-readable string.

Returns: the name of this triangulation.

◆ operator!=()

bool regina::BlockedSFSTriple::operator!= ( const BlockedSFSTriple & other ) const

inline

Determines whether this and the given structure do not represent the same type of blocked sequence of three Seifert fibred spaces.

Specifically, two structures will compare as equal if and only if both structures are formed from the same triple of combinatorial presentations of saturated regions (as returned by the SatRegion comparison operators), presented in the same order, and with their torus boundaries joined using the same pair of 2-by-2 matrices.

This test follows the general rule for most subclasses of StandardTriangulation (excluding fixed structures such as SnappedBall and TriSolidTorus): two objects compare as equal if and only if they have the same combinatorial parameters (which for this subclass is more specific than combinatorial isomorphism, since this test does not account for the many symmetries in a blocked Seifert fibred space).

Parameters

other the structure with which this will be compared.

Returns: true if and only if this and the given structure do not represent the same type of blocked sequence of three Seifert fibred spaces.

◆ operator=() [1/2]

BlockedSFSTriple & regina::BlockedSFSTriple::operator= ( BlockedSFSTriple && src )

defaultnoexcept

Moves the contents of the given structure into this structure.

This is a constant time operation.

The structure that was passed (src) will no longer be usable.

Parameters

src	the structure to move from.

Returns: a reference to this structure.

◆ operator=() [2/2]

BlockedSFSTriple & regina::BlockedSFSTriple::operator= ( const BlockedSFSTriple & src )

default

Sets this to be a copy of the given structure.

This will induce a deep copy of src.

Parameters

src	the structure to copy.

Returns: a reference to this structure.

◆ operator==()

bool regina::BlockedSFSTriple::operator== ( const BlockedSFSTriple & other ) const

inline

Determines whether this and the given structure represent the same type of blocked sequence of three Seifert fibred spaces.

Specifically, two structures will compare as equal if and only if both structures are formed from the same triple of combinatorial presentations of saturated regions (as returned by the SatRegion comparison operators), presented in the same order, and with their torus boundaries joined using the same pair of 2-by-2 matrices.

This test follows the general rule for most subclasses of StandardTriangulation (excluding fixed structures such as SnappedBall and TriSolidTorus): two objects compare as equal if and only if they have the same combinatorial parameters (which for this subclass is more specific than combinatorial isomorphism, since this test does not account for the many symmetries in a blocked Seifert fibred space).

Parameters

other the structure with which this will be compared.

Returns: true if and only if this and the given structure represent the same type of blocked sequence of three Seifert fibred spaces.

◆ recognise() [1/2]

static std::unique_ptr< StandardTriangulation > regina::StandardTriangulation::recognise ( Component< 3 > * component )

staticinherited

Determines whether the given component represents one of the standard triangulations understood by Regina.

The list of recognised triangulations is expected to grow between releases.

If the standard triangulation returned has boundary triangles then the given component must have the same corresponding boundary triangles, i.e., the component cannot have any further identifications of these boundary triangles with each other.

Note that the triangulation-based routine recognise(const Triangulation<3>&) may recognise more triangulations than this routine, since passing an entire triangulation allows access to more information.

Parameters

component the triangulation component under examination.

Returns: the details of the standard triangulation if the given component is recognised, or null otherwise.

◆ recognise() [2/2]

static std::unique_ptr< BlockedSFSTriple > regina::BlockedSFSTriple::recognise ( const Triangulation< 3 > & tri )

static

Determines if the given triangulation is a blocked sequence of three Seifert fibred spaces, as described in the class notes above.

This function returns by (smart) pointer for consistency with StandardTriangulation::recognise(), which makes use of the polymorphic nature of the StandardTriangulation class hierarchy.

Parameters

tri	the triangulation to examine.

Returns: a structure containing details of the blocked triple, or null if the given triangulation is not of this form.

◆ str()

std::string regina::Output< StandardTriangulation , 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.

◆ swap()

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

inlinenoexcept

Swaps the contents of this and the given structure.

Parameters

other the structure whose contents should be swapped with this.

◆ texName()

std::string regina::StandardTriangulation::texName ( ) const

inherited

Returns the name of this specific triangulation 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 name of this triangulation in TeX format.

◆ utf8()

std::string regina::Output< StandardTriangulation , 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::BlockedSFSTriple::writeName ( std::ostream & out ) const

overridevirtual

Writes the name of this triangulation 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::StandardTriangulation.

◆ writeTeXName()

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

overridevirtual

Writes the name of this triangulation 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::StandardTriangulation.

◆ writeTextLong()

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

overridevirtual

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

This may be reimplemented by subclasses, but the parent StandardTriangulation class offers a reasonable default implementation based on writeTextShort().

Python: Not present. Use detail() instead.

Parameters

out	the output stream to which to write.

Reimplemented from regina::StandardTriangulation.

◆ writeTextShort()

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

inlinevirtualinherited

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

This may be reimplemented by subclasses, but the parent StandardTriangulation class offers a reasonable default implementation based on writeName().

Python: Not present. Use str() instead.

Parameters

out	the output stream to which to write.

Reimplemented in regina::LayeredChain, regina::LayeredSolidTorus, regina::SnappedBall, regina::SpiralSolidTorus, and regina::TriSolidTorus.

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

subcomplex/blockedsfstriple.h

Public Member Functions

Static Public Member Functions

Detailed Description

Constructor & Destructor Documentation

◆ BlockedSFSTriple() [1/2]

◆ BlockedSFSTriple() [2/2]

Member Function Documentation

◆ centre()

◆ detail()

◆ end()

◆ homology()

◆ manifold()

◆ matchingReln()

◆ name()

◆ operator!=()

◆ operator=() [1/2]

◆ operator=() [2/2]

◆ operator==()

◆ recognise() [1/2]

◆ recognise() [2/2]

◆ str()

◆ swap()

◆ texName()

◆ utf8()

◆ writeName()

◆ writeTeXName()

◆ writeTextLong()

◆ writeTextShort()