Describes a triangulation of a graph manifold formed by joining a bounded saturated region with a thin I-bundle over the torus, possibly with layerings in between. More...

#include <subcomplex/pluggedtorusbundle.h>

Inheritance diagram for regina::PluggedTorusBundle:

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

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

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

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

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

const TxICore &	bundle () const
	Returns an isomorphic copy of the thin I-bundle that forms part of this triangulation. More...

const Isomorphism< 3 > &	bundleIso () const
	Returns an isomorphism describing how the thin I-bundle forms a subcomplex of this triangulation. More...

const SatRegion &	region () const
	Returns the saturated region that forms part of this triangulation. More...

const Matrix2 &	matchingReln () const
	Returns the matrix describing how the two torus boundaries of the saturated region are joined by the thin I-bundle and layerings. More...

bool	operator== (const PluggedTorusBundle &other) const
	Determines whether this and the given structure represent the same type of plugged torus bundle. More...

bool	operator!= (const PluggedTorusBundle &other) const
	Determines whether this and the given structure do not represent the same type of plugged torus bundle. 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< PluggedTorusBundle >	recognise (const Triangulation< 3 > &tri)
	Determines if the given triangulation is a saturated region joined to a thin I-bundle via optional layerings, 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

Describes a triangulation of a graph manifold formed by joining a bounded saturated region with a thin I-bundle over the torus, possibly with layerings in between.

The thin I-bundle must be untwisted, so that it forms the product T x I with two boundary tori. Moreover, it must be isomorphic to some existing instance of the class TxICore.

The saturated region is described by an object of the class SatRegion. This region must have precisely two boundary annuli. These may be two separate torus boundaries (each formed from its own saturated annulus). Alternatively, the saturated region may have a single boundary formed from both saturated annuli, where this boundary is pinched together so that each annulus becomes its own two-sided torus.

Either way, the saturated region effectively has two torus boundaries, each formed from two triangles of the triangulation. These boundaries are then joined to the two torus boundaries of the thin I-bundle, possibly with layerings in between (for more detail on layerings, see the Layering class). This is illustrated in the following diagram, where the small tunnels show where the torus boundaries are joined (possibly via layerings).

   /--------------------\     /-----------------\
   |                     -----                  |
   |                     -----                  |
   |  Saturated region  |     |  Thin I-bundle  |
   |                     -----                  |
   |                     -----                  |
   --------------------/     -----------------/

The effect of the thin I-bundle and the two layerings is essentially to join the two boundaries of the saturated region according to some non-trivial homeomorphism of the torus. This homeomorphism is specified by a 2-by-2 matrix M as follows.

Suppose that f0 and o0 are directed curves on the first boundary torus and f1 and o1 are directed curves on the second boundary torus, where f0 and f1 represent the fibres of the saturated region and o0 and o1 represent the base orbifold (see the page on Notation for Seifert fibred spaces for terminology). Then the torus boundaries of the saturated region are identified by the thin I-bundle and layerings according to the following relation:

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

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 more detail, 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

◆ PluggedTorusBundle() [1/2]

regina::PluggedTorusBundle::PluggedTorusBundle ( const PluggedTorusBundle & src )

default

Creates a new copy of the given structure.

This will induce a deep copy of src.

Parameters

src	the structure to copy.

◆ PluggedTorusBundle() [2/2]

regina::PluggedTorusBundle::PluggedTorusBundle ( PluggedTorusBundle && src )

defaultnoexcept

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

This is a fast (constant time) operation.

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

Parameters

src	the structure to move from.

Member Function Documentation

◆ bundle()

const TxICore & regina::PluggedTorusBundle::bundle ( ) const

inline

Returns an isomorphic copy of the thin I-bundle that forms part of this triangulation.

Like all objects of class TxICore, the thin I-bundle that is returned is an external object with its own separate triangulation of the product T x I. For information on how the thin I-bundle is embedded within this triangulation, see the routine bundleIso().

Returns: the an isomorphic copy of the thin I-bundle within this triangulation.

◆ bundleIso()

const Isomorphism< 3 > & regina::PluggedTorusBundle::bundleIso ( ) const

inline

Returns an isomorphism describing how the thin I-bundle forms a subcomplex of this triangulation.

The thin I-bundle returned by bundle() does not directly refer to tetrahedra within this triangulation. Instead it contains its own isomorphic copy of the thin I-bundle triangulation (as is usual for objects of class TxICore).

The isomorphism returned by this routine is a mapping from the triangulation bundle().core() to this triangulation, showing how the thin I-bundle appears as a subcomplex of this structure.

Returns: an isomorphism from the thin I-bundle described by bundle() to the tetrahedra of this triangulation.

◆ 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.

◆ 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::PluggedTorusBundle::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::PluggedTorusBundle::matchingReln ( ) const

inline

Returns the matrix describing how the two torus boundaries of the saturated region are joined by the thin I-bundle and layerings.

See the class notes above for details.

Returns: the matching relation between the two region boundaries.

◆ 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::PluggedTorusBundle::operator!= ( const PluggedTorusBundle & other ) const

inline

Determines whether this and the given structure do not represent the same type of plugged torus bundle.

Specifically, two structures will compare as equal if and only if:

both structures use the same type of thin I-bundle with the same parameters (as tested by the TxICore comparison operators);
both structures use saturated regions with the same combinatorial presentation (as tested by the SatRegion comparison operators);
the layerings that connect the thin I-bundle and saturated region in each structure are the same (as tested by the Layering comparison operators), and use the same attaching 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 possible symmetries in a plugged torus bundle).

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 plugged torus bundle.

◆ operator=() [1/2]

PluggedTorusBundle & regina::PluggedTorusBundle::operator= ( const PluggedTorusBundle & 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=() [2/2]

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

defaultnoexcept

Moves the contents of the given structure into this structure.

This is a fast (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==()

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

inline

Determines whether this and the given structure represent the same type of plugged torus bundle.

Specifically, two structures will compare as equal if and only if:

both structures use the same type of thin I-bundle with the same parameters (as tested by the TxICore comparison operators);
both structures use saturated regions with the same combinatorial presentation (as tested by the SatRegion comparison operators);
the layerings that connect the thin I-bundle and saturated region in each structure are the same (as tested by the Layering comparison operators), and use the same attaching 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 possible symmetries in a plugged torus bundle).

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 plugged torus bundle.

◆ 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< PluggedTorusBundle > regina::PluggedTorusBundle::recognise ( const Triangulation< 3 > & tri )

static

Determines if the given triangulation is a saturated region joined to a thin I-bundle via optional layerings, 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: an object containing details of the structure that was found, or null if the given triangulation is not of the form described by this class.

◆ region()

const SatRegion & regina::PluggedTorusBundle::region ( ) const

inline

Returns the saturated region that forms part of this triangulation.

The region refers directly to tetrahedra within this triangulation (as opposed to the thin I-bundle, which refers to a separate external triangulation).

Returns: the saturated region.

◆ 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::PluggedTorusBundle::swap ( PluggedTorusBundle & 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::PluggedTorusBundle::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::PluggedTorusBundle::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::PluggedTorusBundle::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/pluggedtorusbundle.h

Public Member Functions

Static Public Member Functions

Detailed Description

Constructor & Destructor Documentation

◆ PluggedTorusBundle() [1/2]

◆ PluggedTorusBundle() [2/2]

Member Function Documentation

◆ bundle()

◆ bundleIso()

◆ detail()

◆ homology()

◆ manifold()

◆ matchingReln()

◆ name()

◆ operator!=()

◆ operator=() [1/2]

◆ operator=() [2/2]

◆ operator==()

◆ recognise() [1/2]

◆ recognise() [2/2]

◆ region()

◆ str()

◆ swap()

◆ texName()

◆ utf8()

◆ writeName()

◆ writeTeXName()

◆ writeTextLong()

◆ writeTextShort()