Represents an annulus formed from a pair of triangles in a Seifert fibred space. More...

#include <subcomplex/satannulus.h>

Public Member Functions
	SatAnnulus ()
	Creates a new uninitialised structure. More...

	SatAnnulus (const SatAnnulus &)=default
	Creates a new copy of the given structure. More...

	SatAnnulus (const Tetrahedron< 3 > t0, Perm< 4 > r0, const Tetrahedron< 3 > t1, Perm< 4 > r1)
	Creates a new structure initialised to the given values. More...

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

bool	operator== (const SatAnnulus &other) const
	Determines whether or not this and the given structure describe the same specific presentation of a saturated annulus. More...

bool	operator!= (const SatAnnulus &other) const
	Determines whether or not this and the given structure describe different specific presentations of a saturated annulus. More...

unsigned	meetsBoundary () const
	Determines how many triangles of this annulus lie on the boundary of the triangulation. More...

void	switchSides ()
	Converts this into a representation of the same annulus from the other side. More...

SatAnnulus	otherSide () const
	Returns a representation of the same annulus from the other side. More...

void	reflectVertical ()
	Reverses the direction of the vertical fibres in this annulus representation. More...

SatAnnulus	verticalReflection () const
	Returns a representation of this annulus in which the vertical direction of the fibres has been reversed. More...

void	reflectHorizontal ()
	Performs a left-to-right reflection of this annulus representation. More...

SatAnnulus	horizontalReflection () const
	Returns a left-to-right reflected representation of this annulus. More...

void	rotateHalfTurn ()
	Rotates the representation of this annulus by 180 degrees. More...

SatAnnulus	halfTurnRotation () const
	Returns a 180 degree rotated representation of this annulus. More...

std::tuple< bool, bool, bool >	isAdjacent (const SatAnnulus &other) const
	Determines whether this and the given annulus are adjacent, possibly modulo vertical or horizontal reflections. More...

bool	isJoined (const SatAnnulus &other, Matrix2 &matching) const
	Determines whether this and the given annulus are joined in some form, even if the fibres on each annulus are not consistent. More...

bool	isTwoSidedTorus () const
	Determines whether this annulus has its boundaries identified to form an embedded two-sided torus within the surrounding triangulation. More...

void	transform (const Triangulation< 3 > &originalTri, const Isomorphism< 3 > &iso, const Triangulation< 3 > &newTri)
	Adjusts this annulus representation according to the given isomorphism between triangulations. More...

SatAnnulus	image (const Triangulation< 3 > &originalTri, const Isomorphism< 3 > &iso, const Triangulation< 3 > &newTri) const
	Returns the image of this annulus representation under the given isomorphism between triangulations. More...

Static Public Member Functions
static void	attachLST (Tetrahedron< 3 > t0, Perm< 4 > r0, Tetrahedron< 3 > t1, Perm< 4 > r1, long alpha, long beta)
	Attaches a layered solid torus to the given saturated annulus. More...

Public Attributes
const Tetrahedron< 3 > *	tet [2]
	Describes which tetrahedra provide the first and second triangles. More...

Perm< 4 >	roles [2]
	Describes how the first and second triangles match up with individual tetrahedron vertices. More...

Detailed Description

Represents an annulus formed from a pair of triangles in a Seifert fibred space.

This annulus is saturated, i.e., a union of fibres. More than that, the fibres run parallel to the two boundary edges of the annulus.

The annulus is described from one side only. The description includes an array of indices tet[] describing which two tetrahedra provide the triangles of the annulus, as well as an array of permutations roles[] detailing how the annulus matches up with the individual tetrahedron vertices.

The annulus can be drawn as follows, with the upper edge identified with the lower:

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

Suppose that tet[0] and tet[1] are the tetrahedra providing the first and second triangles respectively. Then the markings 0..2 on the first triangle above correspond to vertices roles[0][0..2] of tetrahedron tet[0], and likewise the markings 0..2 on the second triangle above correspond to vertices roles[1][0..2] of tetrahedron tet[1].

Note that the diagram above can also be drawn as follows.

           *--->---*
           | \ 2  1|
   First   |0 \    |  Second
  triangle |   \   | triangle
           |    \ 0|
           |1  2 \ |
           *--->---*

Note also that the labelling of the tetrahedra and their vertices establishes an orientation on the vertical fibres, as well as a left-to-right direction across the annulus.

For convenience we refer to edges roles[][0-1] as vertical, edges roles[][0-2] as horizontal, and edge roles[][1-2] as diagonal. This is illustrated in the following diagrams.

        V  Horizontal       V   Diagonal
        e  *--->---*        e  *--->---*
        r  |   g / |        r  |H\ 2  1|
        t  |  a / 1|        t  | o\    |
        i  | i /   |        i  |  r\   |
        c  |D /    |        c  |   i\ 0|
        a  | / 2  0|        a  |    z\ |
        l  *--->---*        l  *--->---*

These objects are small enough to pass by value and swap with std::swap(), with no need for any specialised move operations or swap functions.

Python: The member arrays tet and roles are accessed for reading through functions tet() and roles() respectively. For instance, the first triangle tetrahedron for the saturated annulus a can be accessed as a.tet(0). These same member arrays are accessed for writing through functions setTet() and setRoles(), so for instance the second triangle vertex roles for the saturated annulus a can be modified by calling a.setRoles(1, newRoles).

Constructor & Destructor Documentation

◆ SatAnnulus() [1/3]

regina::SatAnnulus::SatAnnulus ( )

inline

Creates a new uninitialised structure.

Both tetrahedra will be set to null pointers.

◆ SatAnnulus() [2/3]

regina::SatAnnulus::SatAnnulus ( const SatAnnulus & )

default

Creates a new copy of the given structure.

◆ SatAnnulus() [3/3]

regina::SatAnnulus::SatAnnulus	(	const Tetrahedron< 3 > *	t0,
		Perm< 4 >	r0,
		const Tetrahedron< 3 > *	t1,
		Perm< 4 >	r1
	)

inline

Creates a new structure initialised to the given values.

See the class notes for what the various tetrahedra and permutations mean.

Parameters

t0	the tetrahedron to assign to tet[0].
r0	the permutation to assign to roles[0].
t1	the tetrahedron to assign to tet[1].
r1	the permutation to assign to roles[1].

Member Function Documentation

◆ attachLST()

static void regina::SatAnnulus::attachLST	(	Tetrahedron< 3 > *	t0,
		Perm< 4 >	r0,
		Tetrahedron< 3 > *	t1,
		Perm< 4 >	r1,
		long	alpha,
		long	beta
	)

static

Attaches a layered solid torus to the given saturated annulus.

Instead of passing a SatAnnulus (which only offers const access to the underlying triangluation), you must pass the individual tetrahedra and permutations that describe it.

The layered solid torus will be attached so that the given values alpha and beta describe how the meridinal disc cuts the vertical and horizontal edges of the annulus respectively.

The result will effectively insert an (alpha, beta) exceptional fibre into the Seifert fibred space space, where the vertical edges run parallel to the fibres and the horizontal edges represent the base orbifold. The sign of the fibre is consistent with the fibre inserted by SatLST::adjustSFS() (in particular, negating beta will negate the fibre).

In the case of a (2,1) fibre, the layered solid torus will be degenerate (i.e., the two triangles of the annulus will simply be joined together).

Precondition: The given value alpha is not zero.; The given values alpha and beta are coprime.

Parameters

t0	the tetrahedron corresponding to SatAnnulus::tet[0].
r0	the permutation corresponding to SatAnnulus::roles[0].
t1	the tetrahedron corresponding to SatAnnulus::tet[1].
r1	the permutation corresponding to SatAnnulus::roles[1].
alpha	describes how the meridinal disc of the torus should cut the vertical edges. This may be positive or negative.
beta	describes how the meridinal disc of the torus should cut the horizontal edges. Again this may be positive or negative.

◆ halfTurnRotation()

SatAnnulus regina::SatAnnulus::halfTurnRotation ( ) const

inline

Returns a 180 degree rotated representation of this annulus.

This structure will not be changed. See rotateHalfTurn() for further details.

Returns: a new 180 degree rotation of this annulus.

◆ horizontalReflection()

SatAnnulus regina::SatAnnulus::horizontalReflection ( ) const

inline

Returns a left-to-right reflected representation of this annulus.

This structure will not be changed. See reflectHorizontal() for further details.

Returns: a new left-to-right reflection of this annulus.

◆ image()

SatAnnulus regina::SatAnnulus::image	(	const Triangulation< 3 > &	originalTri,
		const Isomorphism< 3 > &	iso,
		const Triangulation< 3 > &	newTri
	)		const

inline

Returns the image of this annulus representation under the given isomorphism between triangulations.

This annulus representation will not be changed. See transform() for further details.

Parameters

originalTri	the triangulation currently used by this annulus representation.
iso	the mapping from originalTri to newTri.
newTri	the triangulation to be used by the new annulus representation.

◆ isAdjacent()

std::tuple< bool, bool, bool > regina::SatAnnulus::isAdjacent ( const SatAnnulus & other ) const

Determines whether this and the given annulus are adjacent, possibly modulo vertical or horizontal reflections.

That is, this routine determines whether this and the given structure represent opposite sides of the same saturated annulus, where the fibres for both structures are consistent (though possibly reversed). See switchSides() for details on what "opposite sides" means in this context, and see reflectVertical() and reflectHorizontal() for descriptions of the various types of reflection.

Information regarding reflections is returned via the second and third elements of the returned tuple (call these refVert and refHoriz). If the two annuli are identically opposite each other as described by switchSides(), both booleans will be false. If the two annuli are identically opposite after one undergoes a vertical and/or horizontal reflection, then the booleans refVert and/or refHoriz will be true accordingly.

The critical difference between this routine and isJoined() is that this routine insists that the fibres on each annulus be consistent. This routine is thus suitable for examining joins between different sections of the same Seifert fibred space, for example.

Parameters

other the annulus to compare with this.

Returns: a tuple of booleans (adj, refVert, refHoriz), where: adj is true iff some adjacency was found (either with or without reflections); refVert is true iff a vertical reflection is required; and refHoriz is true iff a horizontal reflection is required. If no adjacency was found at all, then both refVert and refHoriz will be false.

◆ isJoined()

bool regina::SatAnnulus::isJoined	(	const SatAnnulus &	other,
		Matrix2 &	matching
	)		const

Determines whether this and the given annulus are joined in some form, even if the fibres on each annulus are not consistent.

This routine treats each annulus as though its boundaries are identified to form a torus (though it does not actually test whether this is true). It then examines whether this and the given annulus represent opposite sides of the same torus. More specifically, it tests whether both annuli are formed from the same pair of triangles, and whether the mapping of 0/1/2 markings from one annulus to the other is the same for each triangle. Note that the triangles are allowed to be switched (i.e., the first triangle of one annulus may be the second triangle of the other).

The critical difference between this routine and isAdjacent() is that this routine allows the fibres on each annulus to be inconsistent. This routine is thus suitable for examining joins between different Seifert fibred blocks in a graph manifold, for example.

If the two annuli are joined, the precise relationship between the curves on each annulus will be returned in the matrix matching. Specifically, let x and y be the oriented curves running from markings 0-1 and 0-2 respectively on the first triangle of this annulus. Likewise, let x' and y' run from markings 0-1 and 0-2 respectively on the first triangle of the annulus other. Then the joining between the two annuli can be expressed as follows:

    [x ]                [x']
    [  ]  =  matching * [  ].
    [y ]                [y']

Parameters

other	the annulus to compare with this.
matching	returns details on how the curves on each annulus are related. If the this and the given annulus are not joined, then this matrix is not touched.

Returns: true if this and the given annulus are found to be joined, or false if they are not.

◆ isTwoSidedTorus()

bool regina::SatAnnulus::isTwoSidedTorus ( ) const

Determines whether this annulus has its boundaries identified to form an embedded two-sided torus within the surrounding triangulation.

It will be verified that:

the two triangles of this annulus are joined along all three pairs of edges to form a torus;
the three edges of this torus remain distinct (i.e., different edges of the torus do not become identified within the larger triangulation);
this torus is two-sided within the surrounding triangulation.

Returns: true if this annulus forms an embedded two-sided torus as described above, or false if it does not.

◆ meetsBoundary()

unsigned regina::SatAnnulus::meetsBoundary ( ) const

Determines how many triangles of this annulus lie on the boundary of the triangulation.

Note that this routine can also be used as a boolean function to determine whether any triangles of the annulus lie on the triangulation boundary.

Returns: the number of triangles of this annulus that lie on the boundary of the triangulation; this will be 0, 1 or 2.

◆ operator!=()

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

inline

Determines whether or not this and the given structure describe different specific presentations of a saturated annulus.

Specifically, in order to compare as equal, two saturated annuli must use the same two numbered tetrahedra, presented in the same order, and with the same roles permutations.

Because this operation compares tetrahedron numbers and not the underlying Tetrahedron objects, it is meaningful to compare saturated annuli from different triangulations.

Parameters

other the structure to compare with this.

Returns: true if and only if both structures describe different specific presentations of a saturated annulus.

◆ operator=()

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

default

Sets this to be a copy of the given structure.

Returns: a reference to this structure.

◆ operator==()

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

inline

Determines whether or not this and the given structure describe the same specific presentation of a saturated annulus.

Specifically, in order to compare as equal, two saturated annuli must use the same two numbered tetrahedra, presented in the same order, and with the same roles permutations.

Because this operation compares tetrahedron numbers and not the underlying Tetrahedron objects, it is meaningful to compare saturated annuli from different triangulations.

Parameters

other the structure to compare with this.

Returns: true if and only if both structures describe the same specific presentation of a saturated annulus.

◆ otherSide()

SatAnnulus regina::SatAnnulus::otherSide ( ) const

inline

Returns a representation of the same annulus from the other side.

This structure will not be changed. See switchSides() for further details.

Precondition: Neither triangle of this annulus is a boundary triangle of the triangulation.

Returns: a new representation of this annulus from the other side.

◆ reflectHorizontal()

void regina::SatAnnulus::reflectHorizontal ( )

inline

Performs a left-to-right reflection of this annulus representation.

The vertical direction of the fibres will remain unchanged, but the first and second triangles will be switched (and the 0..2 markings changed to compensate).

◆ reflectVertical()

void regina::SatAnnulus::reflectVertical ( )

inline

Reverses the direction of the vertical fibres in this annulus representation.

The first and second triangles (as described in the class notes) will remain unchanged, but the markings 0 and 1 on each triangle will be switched.

◆ rotateHalfTurn()

void regina::SatAnnulus::rotateHalfTurn ( )

inline

Rotates the representation of this annulus by 180 degrees.

This has the effect of switching the first and second triangles and also reversing the direction of the vertical fibres.

Calling this routine is equivalent to calling reflectVertical() and then reflectHorizontal().

◆ switchSides()

void regina::SatAnnulus::switchSides ( )

Converts this into a representation of the same annulus from the other side.

The first and second triangles and their 0..2 markings (as described in the class notes) remain unchanged. However, the two tetrahedra that are used to describe the annulus will be replaced by their counterparts on the other side of the annulus (i.e., the two new tetrahedra that meet the two original tetrahedra along the annulus itself).

Precondition: Neither triangle of this annulus is a boundary triangle of the triangulation.

◆ transform()

void regina::SatAnnulus::transform	(	const Triangulation< 3 > &	originalTri,
		const Isomorphism< 3 > &	iso,
		const Triangulation< 3 > &	newTri
	)

Adjusts this annulus representation according to the given isomorphism between triangulations.

The given isomorphism must describe a mapping from originalTri to newTri, and this annulus must refer to tetrahedra in originalTri. This routine will adjust this annulus according to the given isomorphism, so that it refers to the corresponding tetrahedra in newTri (with the roles permutations also updated accordingly).

Precondition: This annulus refers to tetrahedra in originalTri, and iso describes a mapping from originalTri to newTri.

Parameters

originalTri	the triangulation currently used by this annulus representation.
iso	the mapping from originalTri to newTri.
newTri	the triangulation to be used by the updated annulus representation.

◆ verticalReflection()

SatAnnulus regina::SatAnnulus::verticalReflection ( ) const

inline

Returns a representation of this annulus in which the vertical direction of the fibres has been reversed.

This structure will not be changed. See reflectVertical() for further details.

Returns: a new representation of this annulus in which fibres have been reversed.

Member Data Documentation

◆ roles

Perm<4> regina::SatAnnulus::roles[2]

Describes how the first and second triangles match up with individual tetrahedron vertices.

See the class notes for details.

◆ tet

const Tetrahedron<3>* regina::SatAnnulus::tet[2]

Describes which tetrahedra provide the first and second triangles.

See the class notes for details.

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

subcomplex/satannulus.h

Public Member Functions

Static Public Member Functions

Public Attributes

Detailed Description

Constructor & Destructor Documentation

◆ SatAnnulus() [1/3]

◆ SatAnnulus() [2/3]

◆ SatAnnulus() [3/3]

Member Function Documentation

◆ attachLST()

◆ halfTurnRotation()

◆ horizontalReflection()

◆ image()

◆ isAdjacent()

◆ isJoined()

◆ isTwoSidedTorus()

◆ meetsBoundary()

◆ operator!=()

◆ operator=()

◆ operator==()

◆ otherSide()

◆ reflectHorizontal()

◆ reflectVertical()

◆ rotateHalfTurn()

◆ switchSides()

◆ transform()

◆ verticalReflection()

Member Data Documentation

◆ roles

◆ tet