Provides a triangulation of the product T x I (the product of the torus and the interval). More...

#include <subcomplex/txicore.h>

Inheritance diagram for regina::TxICore:

Public Member Functions
virtual	~TxICore ()=default
	Destroys this object. More...

const Triangulation< 3 > &	core () const
	Returns a full copy of the `T x I` triangulation that this object describes. More...

size_t	bdryTet (int whichBdry, int whichTri) const
	Determines which tetrahedron provides the requested boundary triangle. More...

Perm< 4 >	bdryRoles (int whichBdry, int whichTri) const
	Describes which tetrahedron vertices play which roles in the upper and lower boundary triangles. More...

const Matrix2 &	bdryReln (int whichBdry) const
	Returns a 2-by-2 matrix describing the α and β curves on a torus boundary in terms of specific tetrahedron edges. More...

const Matrix2 &	parallelReln () const
	Returns a 2-by-2 matrix describing the parallel relationship between the upper and lower boundary curves. More...

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

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

virtual bool	operator== (const TxICore &other) const =0
	Determines if this and the given `T x I` triangulation are of the same type and have the same parameters. More...

bool	operator!= (const TxICore &other) const
	Determines if this and the given `T x I` triangulation are of different types and/or have different parameters. More...

virtual std::ostream &	writeName (std::ostream &out) const =0
	Writes the name of this specific triangulation of `T x I` to the given output stream. More...

virtual std::ostream &	writeTeXName (std::ostream &out) const =0
	Writes the name of this specific triangulation of `T x I` in TeX format 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...

Protected Member Functions
	TxICore ()=default
	Default constructor that performs no initialisation. More...

	TxICore (const TxICore &src)=default
	Copy constructor. More...

	TxICore (TxICore &&src) noexcept=default
	Move constructor. More...

TxICore &	operator= (const TxICore &)=default
	Copy assignment operator. More...

TxICore &	operator= (TxICore &&) noexcept
	Move assignment operator. More...

void	swapBaseData (TxICore &other) noexcept
	Swaps all data that is managed by this base class with the given triangulation. More...

Protected Attributes
Triangulation< 3 >	core_
	A full copy of the `T x I` triangulation that is described. More...

std::array< std::array< size_t, 2 >, 2 >	bdryTet_
	The tetrahedra that provide the upper and lower boundary triangles. More...

std::array< std::array< Perm< 4 >, 2 >, 2 >	bdryRoles_
	Describes which tetrahedron vertices play which roles in the upper and lower boundary triangles. More...

std::array< Matrix2, 2 >	bdryReln_
	Expresses the α and β curves for each torus boundary in terms of specific tetrahedron edges and vertices. More...

Matrix2	parallelReln_
	Expresses the lower α and β curves in terms of the upper α and β curves. More...

Detailed Description

Provides a triangulation of the product T x I (the product of the torus and the interval).

Generally these triangulations are only one tetrahedron thick (i.e., a "thin I-bundle"), though this is not a strict requirement of this class. Triangulations of this type are generally used as components of larger triangulations (such as layered surface bundles).

This product has two torus boundaries, called the upper and lower boundaries. Each of these boundary tori must be formed from precisely two triangles. This class tracks the mappings between parallel curves on the upper and lower boundaries, as well as mappings from boundary curves to specific tetrahedron edges.

For each of the two torus boundaries, two curves are chosen as generators of the fundamental group; these curves are called α and β. Note that there is no requirement that the upper α and β be parallel to the lower α and β. The parallelReln() routine can be called to establish the precise relationship between these upper and lower curves.

Every object of this class contains a full copy of the triangulation that it describes (so you should not create excessive objects of this class without reason). This triangulation can be accessed through the core() routine.

This is an abstract base class; however, the concrete subclasses offer all of the usual copy, move and swap operations. See each subclass for details.

Constructor & Destructor Documentation

◆ ~TxICore()

virtual regina::TxICore::~TxICore ( )

virtualdefault

Destroys this object.

◆ TxICore() [1/3]

regina::TxICore::TxICore ( )

protecteddefault

Default constructor that performs no initialisation.

◆ TxICore() [2/3]

regina::TxICore::TxICore ( const TxICore & src )

protecteddefault

Copy constructor.

This induces a deep copy of the underlying triangulation.

◆ TxICore() [3/3]

regina::TxICore::TxICore ( TxICore && src )

protecteddefaultnoexcept

Move constructor.

Member Function Documentation

◆ bdryReln()

const Matrix2 & regina::TxICore::bdryReln ( int whichBdry ) const

inline

Returns a 2-by-2 matrix describing the α and β curves on a torus boundary in terms of specific tetrahedron edges.

Consider the first triangle of the given boundary. Let t be the tetrahedron returned by bdryTet(whichBdry, 0) and let p be the permutation returned by bdryRoles(whichBdry, 0).

Let edge01 be the directed edge from vertex p[0] to p[1] of tetrahedron t, and let edge02 be the directed edge from vertex p[0] to p[2] of tetrahedron t. Then the matrix returned by this routine describes how the directed edges edge01 and edge02 relate to the α and β curves on the given boundary. Specifically:

    [ α ]                  [ edge01 ]
    [   ]  =  bdryReln() * [        ] .
    [ β ]                  [ edge02 ]

It is guaranteed that this matrix has determinant +1 or -1.

Parameters

whichBdry 0 if the upper boundary should be examined, or 1 if the lower boundary should be examined.

Returns: the relationship between the boundary curves and tetrahedron edges.

◆ bdryRoles()

Perm< 4 > regina::TxICore::bdryRoles	(	int	whichBdry,
		int	whichTri
	)		const

inline

Describes which tetrahedron vertices play which roles in the upper and lower boundary triangles.

Each boundary torus contains two triangles, whose vertices can be numbered 0, 1 and 2 according to the following diagram. This diagram is completely symmetric, in that edges 1-2 are no more special than edges 0-2 or 0-1. The important observations are that edges 1-2 and 2-1 of each triangle are identified, edges 0-2 and 2-0 of each triangle are identified and edges 0-1 and 1-0 of each triangle are identified.

          *--->>--*
          |0  2 / |
  First   |    / 1|  Second
 triangle v   /   v triangle
          |1 /    |
          | / 2  0|
          *--->>--*

This routine returns a permutation that maps these integers 0,1,2 to real tetrahedron vertices. Let t be the tetrahedron returned by bdryTet(whichBdry, whichTri) and let p be the permutation returned by bdryRoles(whichBdry, whichTri). Then vertices p[0], p[1] and p[2] of tetrahedron t correspond to the markings 0, 1 and 2 respectively in the diagram above (and therefore the boundary triangle is face p[3] of the tetrahedron).

The arguments to this routine affect whether we examine the upper or lower boundary and whether we examine the first or second triangle of this boundary

Parameters

whichBdry	0 if the upper boundary should be examined, or 1 if the lower boundary should be examined.
whichTri	0 if the first boundary triangle should be examined, or 1 if the second boundary triangle should be examined.

Returns: the permutation mapping roles 0, 1 and 2 in the diagram above to real tetrahedron vertex numbers.

◆ bdryTet()

size_t regina::TxICore::bdryTet	(	int	whichBdry,
		int	whichTri
	)		const

inline

Determines which tetrahedron provides the requested boundary triangle.

Recall that the T x I triangulation has two torus boundaries, each consisting of two boundary triangles. This routine returns the specific tetrahedron that provides the given triangle of the given torus boundary.

What is returned is the index number of the tetrahedron within the triangulation. To access the tetrahedron itself, you may call core().tetrahedron(bdryTet(...)).

Note that the same tetrahedron may provide more than one boundary triangle.

Parameters

whichBdry	0 if the upper boundary should be examined, or 1 if the lower boundary should be examined.
whichTri	0 if the first boundary triangle should be examined, or 1 if the second boundary triangle should be examined.

◆ core()

const Triangulation< 3 > & regina::TxICore::core ( ) const

inline

Returns a full copy of the T x I triangulation that this object describes.

Successive calls to this routine will return a reference to the same triangulation (i.e., it is not recreated each time this function is called).

Returns: the full triangulation.

◆ detail()

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

◆ name()

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

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

Returns: the name of this triangulation.

◆ operator!=()

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

inline

Determines if this and the given T x I triangulation are of different types and/or have different parameters.

If this returns false (i.e., both objects compare as equal), then the triangulations returned by core() should also be combinatorially identical.

Parameters

other the T x I triangulation to compare with this.

Returns: true if and only if this and the given triangulation are of different types and/or have different parameters.

◆ operator=() [1/2]

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

protecteddefault

Copy assignment operator.

This is provided here so that subclasses can use it implicitly in their own assignment operators.

◆ operator=() [2/2]

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

inlineprotectednoexcept

Move assignment operator.

This is provided here so that subclasses can use it implicitly in their own assignment operators.

Note: This operator is marked noexcept, even though it calls the Triangulation<3> assignment operator which is not noexcept. This is because the only potential cause of exceptions comes from packet event listeners, and the internal triangulation here does not belong to a packet.

◆ operator==()

virtual bool regina::TxICore::operator== ( const TxICore & other ) const

pure virtual

Determines if this and the given T x I triangulation are of the same type and have the same parameters.

If this returns true, then the triangulations returned by core() should also be combinatorially identical.

Parameters

other the T x I triangulation to compare with this.

Returns: true if and only if this and the given triangulation are of the same type and have the same parameters.

Implemented in regina::TxIDiagonalCore, and regina::TxIParallelCore.

◆ parallelReln()

const Matrix2 & regina::TxICore::parallelReln ( ) const

inline

Returns a 2-by-2 matrix describing the parallel relationship between the upper and lower boundary curves.

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

    [ w  x ]
    [      ] .
    [ y  z ]

In other words, if a_l and b_l are the lower α and β curves respectively, we have

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

Returns: the relationship between the upper and lower boundary curves.

◆ str()

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

◆ swapBaseData()

void regina::TxICore::swapBaseData ( TxICore & other )

protectednoexcept

Swaps all data that is managed by this base class with the given triangulation.

Note: This function is marked noexcept, even though it calls Triangulation<3>::swap() which is not noexcept. This is because the only potential cause of exceptions comes from packet event listeners, and the internal triangulation here does not belong to a packet.

Parameters

other the triangulation whose data should be swapped with this.

◆ texName()

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

Returns the name of this specific triangulation of T x I in TeX format.

No leading or trailing dollar signs will be included.

Returns: the name of this triangulation in TeX format.

◆ utf8()

std::string regina::Output< TxICore , 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()

virtual std::ostream & regina::TxICore::writeName ( std::ostream & out ) const

pure virtual

Writes the name of this specific triangulation of T x I to the given output stream.

The name will be written as a human-readable string.

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.

Implemented in regina::TxIDiagonalCore, and regina::TxIParallelCore.

◆ writeTeXName()

virtual std::ostream & regina::TxICore::writeTeXName ( std::ostream & out ) const

pure virtual

Writes the name of this specific triangulation of T x I in TeX format to the given output stream.

No leading or trailing dollar signs will be written.

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.

Implemented in regina::TxIDiagonalCore, and regina::TxIParallelCore.

◆ writeTextLong()

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

inline

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

Python: Not present. Use detail() instead.

Parameters

out	the output stream to which to write.

◆ writeTextShort()

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

inline

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

Python: Not present. Use str() instead.

Parameters

out	the output stream to which to write.

Member Data Documentation

◆ bdryReln_

std::array<Matrix2, 2> regina::TxICore::bdryReln_

protected

Expresses the α and β curves for each torus boundary in terms of specific tetrahedron edges and vertices.

The elements bdryReln_[0] and bdryReln_[1] refer to the upper and lower boundaries respectively, and each of these matrices must have determinant ±1. See bdryReln() for further details.

◆ bdryRoles_

std::array<std::array<Perm<4>, 2>, 2> regina::TxICore::bdryRoles_

protected

Describes which tetrahedron vertices play which roles in the upper and lower boundary triangles.

See bdryRoles() for details.

◆ bdryTet_

std::array<std::array<size_t, 2>, 2> regina::TxICore::bdryTet_

protected

The tetrahedra that provide the upper and lower boundary triangles.

See bdryTet() for details.

◆ core_

Triangulation<3> regina::TxICore::core_

protected

A full copy of the T x I triangulation that is described.

◆ parallelReln_

Matrix2 regina::TxICore::parallelReln_

protected

Expresses the lower α and β curves in terms of the upper α and β curves.

See parallelReln() for details.

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

subcomplex/txicore.h

Public Member Functions

Protected Member Functions

Protected Attributes

Detailed Description

Constructor & Destructor Documentation

◆ ~TxICore()

◆ TxICore() [1/3]

◆ TxICore() [2/3]

◆ TxICore() [3/3]

Member Function Documentation

◆ bdryReln()

◆ bdryRoles()

◆ bdryTet()

◆ core()

◆ detail()

◆ name()

◆ operator!=()

◆ operator=() [1/2]

◆ operator=() [2/2]

◆ operator==()

◆ parallelReln()

◆ str()

◆ swapBaseData()

◆ texName()

◆ utf8()

◆ writeName()

◆ writeTeXName()

◆ writeTextLong()

◆ writeTextShort()

Member Data Documentation

◆ bdryReln_

◆ bdryRoles_

◆ bdryTet_

◆ core_

◆ parallelReln_