Provides an alternative representation of a single bounded Seifert fibred space. More...

#include <manifold/sfsalt.h>

Inheritance diagram for regina::SFSAlt:

Public Member Functions
	SFSAlt (const SFSpace &original)
	Creates a "basic" alternative representation for the given Seifert fibred space. More...

	SFSAlt (const SFSAlt &base, bool reflect, bool negate=false)
	Creates a new alternative representation from the given alternative representation. More...

	SFSAlt (const SFSAlt &)=default
	Creates a new copy of the given alternative. More...

	SFSAlt (SFSAlt &&) noexcept=default
	Moves the contents of the given alternative into this new alternative. More...

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

SFSAlt &	operator= (SFSAlt &&) noexcept=default
	Moves the contents of the given alternative into this alternative. More...

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

const SFSpace &	alt () const &
	Returns the alternative representation of the original Seifert fibred space. More...

SFSpace &	alt () &
	Returns a non-const reference to the alternative representation of the original Seifert fibred space. More...

SFSpace &&	alt () &&
	Returns an rvalue reference to the alternative representation of the original Seifert fibred space. More...

const Matrix2 &	conversion () const
	Returns the conversion matrix for this alternative space. More...

bool	reflected () const
	Returns whether or not a reflection was used when creating this alternative space. More...

bool	operator== (const SFSAlt &other) const
	Determines whether this and the given alternative representation have identical presentations. More...

bool	operator!= (const SFSAlt &other) const
	Determines whether this and the given alternative representation do not have identical presentations. 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
	A default implementation for detailed output. 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::vector< SFSAlt >	altSet (const SFSpace &sfs)
	Returns a set of alternatives for the given Seifert fibred space. More...

static bool	canNegate (const SFSpace &sfs)
	Determines whether the SFSAlt class constructor will be able to negate all exceptional fibres without reflecting the underlying Seifert fibred space. More...

Detailed Description

Provides an alternative representation of a single bounded Seifert fibred space.

Such alternatives are made possible by altering the curves made by the fibre and base orbifold on a boundary torus.

This class is designed to help in finding simple representations of graph manifolds (or, indeed, any non-geometric manifolds containing Seifert fibred blocks).

Each alternative comes with its own representation of the original Seifert fibred space, along with instructions for converting fibre/base curves on the boundary tori between the original and alternative spaces.

The alternative representation will generally be as simple as possible (and indeed the hope is that it will be simpler than the original). In particular, each alternative space is guaranteed to have obstruction constant zero. The base orbifold may be changed entirely (for instance, an orientable Seifert fibred space over the Mobius band with no exceptional fibres will be converted to a Seifert fibred space over the disc with two exceptional fibres).

The conversions between boundary curves are described by a conversion matrix M as follows. Consider the first boundary torus. Let f_old and o_old be directed curves on this boundary representing the fibre and base orbifold of the original space, and let f_alt and o_alt be directed curves on this same boundary representing the fibre and base orbifold of the new alternative space. Then

    [f_alt]         [f_old]
    [     ]  =  M * [     ].
    [o_alt]         [o_old]

Note that this only applies to the first boundary torus! If the Seifert fibred space has more than one boundary, then for the remaining boundaries the unoriented fibre and base curves remain the same. More specifically, the directed fibre remains identical, and the directed curve representing the base orbifold is reversed if and only if a reflection was used in creating the alternative space, as returned by reflected().

See the page on Notation for Seifert fibred spaces for details on some of the terminology used above.

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 SFSAlt still requires a non-trivial (but constant sized) amount of data to be copied even in a move operation.

Constructor & Destructor Documentation

◆ SFSAlt() [1/4]

regina::SFSAlt::SFSAlt ( const SFSpace & original )

Creates a "basic" alternative representation for the given Seifert fibred space.

This will:

reduce the parameters via SFSpace::reduce(), without reflecting;
add twists to the first boundary torus so that the obstruction constant b in the Seifert fibred space becomes zero;
if the Seifert fibred space is over M/n2 with no exceptional fibres (where M represents the Mobius band), it will switch the fibre and orbifold curves to give a Seifert fibred space over the disc with two exception fibres.

Precondition: The given Seifert fibred space has at least one torus boundary.

Parameters

original the original Seifert fibred space for which we are creating a set of alternative representations.

◆ SFSAlt() [2/4]

regina::SFSAlt::SFSAlt	(	const SFSAlt &	base,
		bool	reflect,
		bool	negate = `false`
	)

Creates a new alternative representation from the given alternative representation.

The changes will include:

if reflect is true, then this will reflect the Seifert fibred space;
if negate is true, then this will attempt to replace all fibres (alpha, beta) with (alpha, alpha - beta), without any further reflections (but see below for conditions on when this is possible);
in all cases, this will reduce the parameters via SFSpace::reduce() and add twists to give obstruction constant zero.

Asking for reflection is always valid. However, the option to negate without reflection is possible only if:

it is possible to negate an exceptional fibre by sliding it around the base orbifold (as in SFSpace::fibreNegating()); and
the number of exceptional fibres is odd.

For convenience, you can test these conditions using the static function canNegate().

Precondition: If negate is true, then the underlying Seifert fibred space satisfies the requirements outlined above, or equivalently, canNegate(base.sfs()) is true.

Note: This constructor only makes sense if at least one of reflect or negate is true, since base should already be in a reduced form with zero obstruction constant.

Parameters

base	the Seifert fibred space representation that will be used as a starting point for this new alternative.
reflect	`true` if we should reflect the Seifert fibred space.
negate	`true` if we should attempt to negate all exceptional fibres without reflecting, as described above.

◆ SFSAlt() [3/4]

regina::SFSAlt::SFSAlt ( const SFSAlt & )

default

Creates a new copy of the given alternative.

◆ SFSAlt() [4/4]

regina::SFSAlt::SFSAlt ( SFSAlt && )

defaultnoexcept

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

This is a fast (constant time) operation.

The alternative that was passed will no longer be usable.

Member Function Documentation

◆ alt() [1/3]

SFSpace & regina::SFSAlt::alt ( ) &

inline

Returns a non-const reference to the alternative representation of the original Seifert fibred space.

Returns: the alternative representation.

◆ alt() [2/3]

SFSpace && regina::SFSAlt::alt ( ) &&

inline

Returns an rvalue reference to the alternative representation of the original Seifert fibred space.

Returns: the alternative representation.

◆ alt() [3/3]

const SFSpace & regina::SFSAlt::alt ( ) const &

inline

Returns the alternative representation of the original Seifert fibred space.

Returns: the alternative representation.

◆ altSet()

std::vector< SFSAlt > regina::SFSAlt::altSet ( const SFSpace & sfs )

inlinestatic

Returns a set of alternatives for the given Seifert fibred space.

These will consist of (1) the alternative base created by passing the original Seifert fibred space to the SFSAlt constructor; and (2) all alternatives created by passing base with appropriate reflection and negation arguments to the SFSAlt constructor. The combinations of reflection and negation arguments that are used (and hence the size of the set that is returned) will depend on the properties of the original space.

Note that in general, none of the alternatives will have a representation identical to the original (generally these alternative representations will be simpler if possible).

It is guaranteed that the set that is returned will be non-empty.

Precondition: The given Seifert fibred space has at least one torus boundary.

Parameters

sfs	the original Seifert fibred space.

Returns: the resulting set of alternative representations for sfs.

◆ canNegate()

bool regina::SFSAlt::canNegate ( const SFSpace & sfs )

inlinestatic

Determines whether the SFSAlt class constructor will be able to negate all exceptional fibres without reflecting the underlying Seifert fibred space.

See the constructor SFSAlt(const SFSAlt&, bool, bool) for further details.

Precondition: The given Seifert fibred space has at least one torus boundary.

Parameters

sfs	the Seifert fibred space that we are attempting to represent.

Returns: true if and only if it is possible to set the negation argument to true in the SFSAlt class constructor.

◆ conversion()

const Matrix2 & regina::SFSAlt::conversion ( ) const

inline

Returns the conversion matrix for this alternative space.

This matrix describes the fibre and base curves of the alternative space on the first boundary torus in terms of the fibre and base curves of the original space (which was passed to the SFSAlt constructor). See the class notes above for details.

Note that this conversion matrix applies only to the first boundary torus! If there is more than one boundary, the remaining boundary conversions are simpler and depend only on whether a reflection has been used or not. See reflected() or the class notes for details.

Returns: the conversion matrix for this alternative space.

◆ detail()

template<class T , bool supportsUtf8 = false>

std::string regina::Output< T, supportsUtf8 >::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.

◆ operator!=()

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

inline

Determines whether this and the given alternative representation do not have identical presentations.

To be considered identical, the two alternatives must have equal alternative SFSpace representations, equal conversion matrices, and either both must have used a reflection or both must have not used a reflection. In other words, this is equivalent to testing all of alt(), conversion() and reflected() for equality.

Parameters

other the alternative to compare against this.

Returns: true if and only if this and the given alternative do not have identical presentations, as described above.

◆ operator=() [1/2]

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

default

Sets this to be a copy of the given alternative.

Returns: a reference to this alternative.

◆ operator=() [2/2]

SFSAlt & regina::SFSAlt::operator= ( SFSAlt && )

defaultnoexcept

Moves the contents of the given alternative into this alternative.

This is a fast (constant time) operation.

The alternative that was passed will no longer be usable.

Returns: a reference to this alternative.

◆ operator==()

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

inline

Determines whether this and the given alternative representation have identical presentations.

To be considered identical, the two alternatives must have equal alternative SFSpace representations, equal conversion matrices, and either both must have used a reflection or both must have not used a reflection. In other words, this is equivalent to testing all of alt(), conversion() and reflected() for equality.

Parameters

other the alternative to compare against this.

Returns: true if and only if this and the given alternative have identical presentations, as described above.

◆ reflected()

bool regina::SFSAlt::reflected ( ) const

inline

Returns whether or not a reflection was used when creating this alternative space.

This determines the conversion between boundary curves for all boundary tori after the first.

More specifically, if no reflection was used then the directed fibre and base curves are identical for the original and alternative spaces. If a reflection was used, then the directed fibres are identical but the directed base curves are reversed.

The conversion between curves on the first boundary torus is generally more complex, and is returned as a matrix by the conversion() routine.

You can also test whether a reflection was used by examining whether the conversion matrix has determinant 1 or -1. However, calling reflected() is both simpler and a little faster.

Returns: true if a reflection was used in creating this alternative space, or false if no reflection was used.

◆ str()

template<class T , bool supportsUtf8 = false>

std::string regina::Output< T, supportsUtf8 >::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::SFSAlt::swap ( SFSAlt & other )

inlinenoexcept

Swaps the contents of this and the given alternative.

Parameters

other the alternative whose contents should be swapped with this.

◆ utf8()

template<class T , bool supportsUtf8 = false>

std::string regina::Output< T, supportsUtf8 >::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.

◆ writeTextLong()

void regina::ShortOutput< SFSAlt , false >::writeTextLong ( std::ostream & out ) const

inlineinherited

A default implementation for detailed output.

This routine simply calls T::writeTextShort() and appends a final newline.

Python: Not present; instead you can call detail() from the subclass T, which returns this output as a string.

Parameters

out	the output stream to which to write.

◆ writeTextShort()

void regina::SFSAlt::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.

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

manifold/sfsalt.h

Public Member Functions

Static Public Member Functions

Detailed Description

Constructor & Destructor Documentation

◆ SFSAlt() [1/4]

◆ SFSAlt() [2/4]

◆ SFSAlt() [3/4]

◆ SFSAlt() [4/4]

Member Function Documentation

◆ alt() [1/3]

◆ alt() [2/3]

◆ alt() [3/3]

◆ altSet()

◆ canNegate()

◆ conversion()

◆ detail()

◆ operator!=()

◆ operator=() [1/2]

◆ operator=() [2/2]

◆ operator==()

◆ reflected()

◆ str()

◆ swap()

◆ utf8()

◆ writeTextLong()

◆ writeTextShort()