Describes a layered torus bundle. More...

#include <subcomplex/layeredtorusbundle.h>

Inheritance diagram for regina::LayeredTorusBundle:

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

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

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

const TxICore &	core () const
	Returns the `T x I` triangulation at the core of this layered torus bundle. More...

const Isomorphism< 3 > &	coreIso () const
	Returns the isomorphism describing how the core `T x I` appears as a subcomplex of this layered torus bundle. More...

const Matrix2 &	layeringReln () const
	Returns a 2-by-2 matrix describing how the layering of tetrahedra relates curves on the two torus boundaries of the core `T x I`. More...

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

bool	operator!= (const LayeredTorusBundle &other) const
	Determines whether this and the given structure represent different types of layered 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...

AbelianGroup	homology () const override
	Returns the expected first homology group of this triangulation, if such a 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 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< LayeredTorusBundle >	recognise (const Triangulation< 3 > &tri)
	Determines if the given triangulation is a layered torus bundle. 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 layered torus bundle.

This is a triangulation of a torus bundle over the circle formed as follows.

We begin with a thin I-bundle over the torus, i.e,. a triangulation of the product T x I that is only one tetrahedron thick. This is referred to as the core, and is described by an object of type TxICore.

We then identify the upper and lower torus boundaries of this core according to some homeomorphism of the torus. This may be impossible due to incompatible boundary edges, and so we allow a layering of tetrahedra over one of the boundari tori in order to adjust the boundary edges accordingly. Layerings are described in more detail in the Layering class.

Given the parameters of the core T x I and the specific layering, the monodromy for this torus bundle over the circle can be calculated. The manifold() routine returns details of the corresponding 3-manifold.

All optional StandardTriangulation routines are implemented for this class.

This class supports copying but does not implement separate move operations, since its internal data is so small that copying is just as efficient. It implements the C++ Swappable requirement via its own member and global swap() functions, for consistency with the other StandardTriangulation subclasses. Note that the only way to create these objects (aside from copying or moving) is via the static member function recognise().

Constructor & Destructor Documentation

◆ LayeredTorusBundle()

regina::LayeredTorusBundle::LayeredTorusBundle ( const LayeredTorusBundle & )

default

Creates a new copy of the given structure.

Member Function Documentation

◆ core()

const TxICore & regina::LayeredTorusBundle::core ( ) const

inline

Returns the T x I triangulation at the core of this layered torus bundle.

This is the product T x I whose boundaries are joined (possibly via some layering of tetrahedra).

Note that the triangulation returned by TxICore::core() (that is, LayeredTorusBundle::core().core()) may well use different tetrahedron and vertex numbers. That is, an isomorphic copy of it appears within this layered surface bundle but the individual tetrahedra and vertices may have been permuted. For a precise mapping from the TxICore::core() triangulation to this triangulation, see the routine coreIso().

Returns: the core T x I triangulation.

◆ coreIso()

const Isomorphism< 3 > & regina::LayeredTorusBundle::coreIso ( ) const

inline

Returns the isomorphism describing how the core T x I appears as a subcomplex of this layered torus bundle.

As described in the core() notes, the core T x I triangulation returned by TxICore::core() appears within this layered torus bundle, but not necessarily with the same tetrahedron or vertex numbers.

This routine returns an isomorphism that maps the tetrahedra and vertices of the core T x I triangulation (as returned by LayeredTorusBundle::core().core()) to the tetrahedra and vertices of this overall layered torus bundle.

Returns: the isomorphism from the core T x I to this layered torus bundle.

◆ 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()

AbelianGroup regina::LayeredTorusBundle::homology ( ) const

overridevirtual

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 from regina::StandardTriangulation.

◆ layeringReln()

const Matrix2 & regina::LayeredTorusBundle::layeringReln ( ) const

inline

Returns a 2-by-2 matrix describing how the layering of tetrahedra relates curves on the two torus boundaries of the core T x I.

The TxICore class documentation describes generating α and β curves on the two torus boundaries of the core T x I (which are referred to as the upper and lower boundaries). The two boundary tori are parallel in two directions: through the core, and through the layering. It is desirable to know the parallel relationship between the two sets of boundary curves in each direction.

The relationship through the core is already described by TxICore::parallelReln(). This routine describes the relationship through the layering.

Let a_u and b_u be the α and β curves on the upper boundary torus, and let a_l and b_l be the α and β curves on the lower boundary torus. Suppose that the upper α is parallel to w.a_l + x.b_l, and that the upper β is parallel to y.a_l + z.b_l. Then the matrix returned will be

    [ w  x ]
    [      ] .
    [ y  z ]

In other words,

    [ a_u ]                      [ a_l ]
    [     ]  =  layeringReln() * [     ] .
    [ b_u ]                      [ b_l ]

It can be observed that this matrix expresses the upper boundary curves in terms of the lower, whereas TxICore::parallelReln() expresses the lower boundary curves in terms of the upper. This means that the monodromy describing the overall torus bundle over the circle can be calculated as

    M  =  layeringReln() * core().parallelReln()

or alternatively using the similar matrix

    M'  =  core().parallelReln() * layeringReln() .

Note that in the degenerate case where there is no layering at all, this matrix is still perfectly well defined; in this case it describes a direct identification between the upper and lower boundary tori.

Returns: the relationship through the layering between the upper and lower boundary curves of the core T x I.

◆ manifold()

std::unique_ptr< Manifold > regina::LayeredTorusBundle::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.

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

inline

Determines whether this and the given structure represent different types of layered torus bundle.

Specifically, two layered torus bundles will compare as equal if and only if their core T x I triangulations have the same combinatorial parameters, and their layering relations are the same.

In particular, if you invert a layered torus bundle (which means the layering relation becomes its inverse matrix), the resulting layered torus bundle will generally not compare as equal.

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 recognise inversion and also does not recognise symmetries within the T x I core).

Parameters

other the structure with which this will be compared.

Returns: true if and only if this and the given structure represent different types of layered torus bundle.

◆ operator=()

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

default

Sets this to be a copy of the given structure.

Returns: a reference to this structure.

◆ operator==()

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

inline

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

Specifically, two layered torus bundles will compare as equal if and only if their core T x I triangulations have the same combinatorial parameters, and their layering relations are the same.

In particular, if you invert a layered torus bundle (which means the layering relation becomes its inverse matrix), the resulting layered torus bundle will generally not compare as equal.

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 recognise inversion and also does not recognise symmetries within the T x I core).

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 layered 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< LayeredTorusBundle > regina::LayeredTorusBundle::recognise ( const Triangulation< 3 > & tri )

static

Determines if the given triangulation is a layered torus bundle.

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 layered torus bundle, or null if the given triangulation is not a layered torus bundle.

◆ 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::LayeredTorusBundle::swap ( LayeredTorusBundle & 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::LayeredTorusBundle::writeName ( std::ostream & out ) const

inlineoverridevirtual

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::LayeredTorusBundle::writeTeXName ( std::ostream & out ) const

inlineoverridevirtual

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::LayeredTorusBundle::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/layeredtorusbundle.h

Public Member Functions

Static Public Member Functions

Detailed Description

Constructor & Destructor Documentation

◆ LayeredTorusBundle()

Member Function Documentation

◆ core()

◆ coreIso()

◆ detail()

◆ homology()

◆ layeringReln()

◆ manifold()

◆ name()

◆ operator!=()

◆ operator=()

◆ operator==()

◆ recognise() [1/2]

◆ recognise() [2/2]

◆ str()

◆ swap()

◆ texName()

◆ utf8()

◆ writeName()

◆ writeTeXName()

◆ writeTextLong()

◆ writeTextShort()