Regina Calculation Engine
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regina::NSVectorQuadClosed Class Referenceabstract


The vector for a normal surface in an ideal triangulation, expressed using quad coordinates and enumerated to include closed surfaces only. More...

#include <surfaces/nsvectorquadclosed.h>

Inheritance diagram for regina::NSVectorQuadClosed:
regina::NSVectorQuad regina::NSVectorMirrored regina::NormalSurfaceVector

Public Member Functions

 NSVectorQuadClosed (size_t length)
 Creates a new vector all of whose entries are initialised to zero. More...
 
 NSVectorQuadClosed (const Vector< LargeInteger > &cloneMe)
 Creates a new vector that is a clone of the given vector. More...
 
virtual NormalSurfaceVectormakeMirror (const Triangulation< 3 > *triang) const override
 Creates a new mirror vector corresponding to this vector. More...
 
virtual const Vertex< 3 > * isVertexLink (const Triangulation< 3 > *triang) const override
 Determines if a rational multiple of the normal surface represented is the link of a single vertex. More...
 
virtual LargeInteger octs (size_t tetIndex, int octType, const Triangulation< 3 > *triang) const override
 Returns the number of octagonal discs of the given type in this normal surface. More...
 
virtual void setElement (size_t index, const LargeInteger &value) override
 Sets the given normal coordinate to the given value. More...
 
virtual void operator+= (const NormalSurfaceVector &other) override
 Adds the given vector to this vector. More...
 
virtual void scaleDown () override
 Scales this vector down by the greatest common divisor of all its elements. More...
 
virtual LargeInteger triangles (size_t tetIndex, int vertex, const Triangulation< 3 > *triang) const override
 Returns the number of triangular discs of the given type in this normal surface. More...
 
virtual LargeInteger orientedTriangles (size_t tetIndex, int vertex, const Triangulation< 3 > *triang, bool orientation) const override
 Returns the number of oriented triangular discs of the given type in this normal surface. More...
 
virtual LargeInteger quads (size_t tetIndex, int quadType, const Triangulation< 3 > *triang) const override
 Returns the number of quadrilateral discs of the given type in this normal surface. More...
 
virtual LargeInteger orientedQuads (size_t tetIndex, int quadType, const Triangulation< 3 > *triang, bool orientation) const override
 Returns the number of oriented quadrilateral discs of the given type in this normal surface. More...
 
virtual LargeInteger edgeWeight (size_t edgeIndex, const Triangulation< 3 > *triang) const override
 Returns the number of times this normal surface crosses the given edge. More...
 
virtual LargeInteger arcs (size_t triIndex, int triVertex, const Triangulation< 3 > *triang) const override
 Returns the number of arcs in which this normal surface intersects the given triangle in the given direction. More...
 
const Raycoords () const
 Gives read-only access to the underlying vector of coordinates. More...
 
virtual NormalSurfaceVectorclone () const =0
 Creates a newly allocated clone of this vector. More...
 
size_t size () const
 Returns the number of coordinates in the underlying vector. More...
 
const LargeIntegeroperator[] (size_t index) const
 Returns the given coordinate from the underlying vector. More...
 
virtual bool allowsAlmostNormal () const =0
 Determines if the specific underlying coordinate system allows for almost normal surfaces, that is, allows for octagonal discs. More...
 
virtual bool allowsSpun () const =0
 Determines if the specific underlying coordinate system allows for spun-normal surfaces; that is, surfaces with infinitely many triangles. More...
 
virtual bool allowsOriented () const =0
 Determines if the specific underlying coordinate system allows for transversely oriented normal surfaces. More...
 
virtual bool hasMultipleOctDiscs (const Triangulation< 3 > *triang) const
 Determines if this normal surface has more than one octagonal disc. More...
 
virtual bool isCompact (const Triangulation< 3 > *triang) const
 Determines if the normal surface represented is compact (has finitely many discs). More...
 
virtual bool isVertexLinking (const Triangulation< 3 > *triang) const
 Determines if the normal surface represented is vertex linking. More...
 
virtual std::pair< const Edge< 3 > *, const Edge< 3 > * > isThinEdgeLink (const Triangulation< 3 > *triang) const
 Determines if a rational multiple of the normal surface represented is the thin link of a single edge. More...
 
virtual bool isSplitting (const Triangulation< 3 > *triang) const
 Determines if the normal surface represented is a splitting surface in the given triangulation. More...
 
virtual LargeInteger isCentral (const Triangulation< 3 > *triang) const
 Determines if the normal surface represented is a central surface in the given triangulation. More...
 

Static Public Member Functions

static NormalSurfaceVectormakeZeroVector (const Triangulation< 3 > *triangulation)
 
static MatrixIntmakeMatchingEquations (const Triangulation< 3 > *triangulation)
 
static NormalSurfaceVectormakeMirror (const Ray &original, const Triangulation< 3 > *triang)
 
static EnumConstraintsmakeEmbeddedConstraints (const Triangulation< 3 > *triangulation)
 

Protected Attributes

Ray coords_
 The raw vector of normal coordinates. More...
 

Detailed Description


The vector for a normal surface in an ideal triangulation, expressed using quad coordinates and enumerated to include closed surfaces only.

The underlying coordinate system is identical to "plain" quad coordinates, as represented by the parent class NSVectorQuad. The difference with NSVectorQuadClosed is that, when enumerating surfaces in this class, it adds linear constraints to ensure that only closed normal surfaces are obtained (as opposed to spun-normal surfaces). The resulting solution space is the space Q_0 as described in "Computing closed essential surfaces in knot complements", by Burton, Coward and Tillmann, in SCG ’13: Proceedings of the 29th Annual Symposium on Computational Geometry, ACM, 2013, pp. 405–414.

Note that, if a vertex surface in quad coordinates is closed, then it will also be a vertex surface in this system of "closed quad coordinates". However, the converse is not true: a vertex surface in closed quad coordinates need not be a vertex in "plain" quad coordinates.

Recall that, in quad coordinates, if there are t tetrahedra in the underlying triangulation then there are precisely 3t coordinates. The first three coordinates will be for the first tetrahedron, the next three for the second tetrahedron and so on. For each tetrahedron, the three individual coordinates represent the number of quadrilateral discs of type 0, 1 and 2 (see NormalSurface::quads()).

Precondition
This coordinate system can only be used with a limited class of triangulations, as documented in the coordinate constant NS_QUAD_CLOSED. Currently, such triangulations must be oriented and ideal, with precisely one torus cusp and no other boundary components or internal vertices. Moreover, SnapPea must be able to work with them without retriangulating (though this should follow from the other constraints).
Python
Not present.

Constructor & Destructor Documentation

◆ NSVectorQuadClosed() [1/2]

regina::NSVectorQuadClosed::NSVectorQuadClosed ( size_t  length)
inline

Creates a new vector all of whose entries are initialised to zero.

Parameters
lengththe number of elements in the new vector.

◆ NSVectorQuadClosed() [2/2]

regina::NSVectorQuadClosed::NSVectorQuadClosed ( const Vector< LargeInteger > &  cloneMe)
inline

Creates a new vector that is a clone of the given vector.

Parameters
cloneMethe vector to clone.

Member Function Documentation

◆ allowsAlmostNormal()

virtual bool regina::NormalSurfaceVector::allowsAlmostNormal ( ) const
pure virtualinherited

Determines if the specific underlying coordinate system allows for almost normal surfaces, that is, allows for octagonal discs.

Note that this has nothing to do with whether or not this specific surface contains octagonal discs.

Returns
true if and only if almost normal surfaces are allowed.

◆ allowsOriented()

virtual bool regina::NormalSurfaceVector::allowsOriented ( ) const
pure virtualinherited

Determines if the specific underlying coordinate system allows for transversely oriented normal surfaces.

Returns
true if and only if transverse orientations are supported.

◆ allowsSpun()

virtual bool regina::NormalSurfaceVector::allowsSpun ( ) const
pure virtualinherited

Determines if the specific underlying coordinate system allows for spun-normal surfaces; that is, surfaces with infinitely many triangles.

Note that this has nothing to do with whether or not this specific surface contains infinitely many triangles.

Returns
true if and only if spun-normal surfaces are allowed.

◆ arcs()

LargeInteger regina::NSVectorMirrored::arcs ( size_t  triIndex,
int  triVertex,
const Triangulation< 3 > *  triang 
) const
inlineoverridevirtualinherited

Returns the number of arcs in which this normal surface intersects the given triangle in the given direction.

See NormalSurface::arcs() for further details.

Parameters
triIndexthe index in the triangulation of the triangle in which we are interested; this should be between 0 and Triangulation<3>::countTriangles()-1 inclusive.
triVertexthe vertex of the triangle (0, 1 or 2) around which the arcs of intersection that we are interested in lie; only these arcs will be counted.
triangthe triangulation in which this normal surface lives.
Returns
the number of times this normal surface intersect the given triangle with the given arc type.

Implements regina::NormalSurfaceVector.

◆ clone()

virtual NormalSurfaceVector* regina::NormalSurfaceVector::clone ( ) const
pure virtualinherited

Creates a newly allocated clone of this vector.

The clone will be of the same subclass of NormalSurfaceVector as this vector.

◆ coords()

const Ray & regina::NormalSurfaceVector::coords ( ) const
inlineinherited

Gives read-only access to the underlying vector of coordinates.

Returns
the vector of coordinates.

◆ edgeWeight()

LargeInteger regina::NSVectorMirrored::edgeWeight ( size_t  edgeIndex,
const Triangulation< 3 > *  triang 
) const
inlineoverridevirtualinherited

Returns the number of times this normal surface crosses the given edge.

See NormalSurface::edgeWeight() for further details.

Parameters
edgeIndexthe index in the triangulation of the edge in which we are interested; this should be between 0 and Triangulation<3>::countEdges()-1 inclusive.
triangthe triangulation in which this normal surface lives.
Returns
the number of times this normal surface crosses the given edge.

Implements regina::NormalSurfaceVector.

◆ hasMultipleOctDiscs()

virtual bool regina::NormalSurfaceVector::hasMultipleOctDiscs ( const Triangulation< 3 > *  triang) const
virtualinherited

Determines if this normal surface has more than one octagonal disc.

It may be assumed that at most one octagonal disc type exists in this surface. This routine will return true if an octagonal type does exist and its coordinate is greater than one.

The default implementation for this routine simply calculates all the octagonal coordinates and returns as soon as a positive or negative result can be established. Subclasses of NormalSurfaceVector should override this if they can provide a faster implementation.

If a subclass does not allow for almost normal surfaces, this routine will never be called and thus does not need to be overwritten.

Precondition
At most one octagonal disc type exists in this surface.
This normal surface vector is using a coordinate system that allows for almost normal surfaces.
Parameters
triangthe triangulation in which this normal surface lives.
Returns
true if and only if there is an octagonal disc type present and its coordinate is greater than one.

◆ isCentral()

virtual LargeInteger regina::NormalSurfaceVector::isCentral ( const Triangulation< 3 > *  triang) const
virtualinherited

Determines if the normal surface represented is a central surface in the given triangulation.

A central surface is a compact surface containing at most one normal or almost normal disc per tetrahedron. If the surface is central, the number of tetrahedra it meets (i.e., the number of discs in the surface) will be returned.

The default implementation for this routine simply runs through and checks the count for each disc type. Subclasses of NormalSurfaceVector should override this if they can provide a faster implementation.

Parameters
triangthe triangulation in which this normal surface lives.
Returns
the number of tetrahedra that the surface meets if it is a central surface, or 0 if it is not a central surface.

◆ isCompact()

virtual bool regina::NormalSurfaceVector::isCompact ( const Triangulation< 3 > *  triang) const
virtualinherited

Determines if the normal surface represented is compact (has finitely many discs).

The default implementation for this routine simply runs through every disc type until a disc type with infinite disc count is found or all disc types have been examined. Subclasses of NormalSurfaceVector should override this if they can provide a faster implementation.

Parameters
triangthe triangulation in which this normal surface lives.
Returns
true if and only if the normal surface represented is compact.

◆ isSplitting()

virtual bool regina::NormalSurfaceVector::isSplitting ( const Triangulation< 3 > *  triang) const
virtualinherited

Determines if the normal surface represented is a splitting surface in the given triangulation.

A splitting surface is a compact surface containing precisely one quad per tetrahedron and no other normal (or almost normal) discs.

The default implementation for this routine simply runs through and checks the count for each disc type. Subclasses of NormalSurfaceVector should override this if they can provide a faster implementation.

Parameters
triangthe triangulation in which this normal surface lives.
Returns
true if and only if the normal surface represented is a splitting surface.

◆ isThinEdgeLink()

virtual std::pair<const Edge<3>*, const Edge<3>*> regina::NormalSurfaceVector::isThinEdgeLink ( const Triangulation< 3 > *  triang) const
virtualinherited

Determines if a rational multiple of the normal surface represented is the thin link of a single edge.

If there are two different edges e1 and e2 for which the surface could be expressed as the thin link of either e1 or e2, the pair (e1,e2) will be returned. If the surface is the thin link of only one edge e, the pair (e,0) will be returned. If the surface is not the thin link of any edges, the pair (0,0) will be returned.

The default implementation for this routine involves counting the number of discs of every type. Subclasses of NormalSurfaceVector should override this if they can provide a faster implementation.

Parameters
triangthe triangulation in which this normal surface lives.
Returns
a pair containing the edge(s) linked by this surface, as described above.

◆ isVertexLink()

const Vertex< 3 > * regina::NSVectorQuad::isVertexLink ( const Triangulation< 3 > *  triang) const
inlineoverridevirtualinherited

Determines if a rational multiple of the normal surface represented is the link of a single vertex.

The default implementation for this routine involves counting the number of discs of every type. Subclasses of NormalSurfaceVector should override this if they can provide a faster implementation.

Parameters
triangthe triangulation in which this normal surface lives.
Returns
the vertex linked by this surface, or 0 if this surface is not the link of a single vertex.

Reimplemented from regina::NormalSurfaceVector.

◆ isVertexLinking()

virtual bool regina::NormalSurfaceVector::isVertexLinking ( const Triangulation< 3 > *  triang) const
virtualinherited

Determines if the normal surface represented is vertex linking.

A vertex linking surface contains only triangles.

The default implementation for this routine simply runs through every non-triangular disc type ensuring that each has no corresponding discs. Subclasses of NormalSurfaceVector should override this if they can provide a faster implementation.

Parameters
triangthe triangulation in which this normal surface lives.
Returns
true if and only if the normal surface represented is vertex linking.

◆ makeMirror()

NormalSurfaceVector * regina::NSVectorQuad::makeMirror ( const Triangulation< 3 > *  triang) const
inlineoverridevirtualinherited

Creates a new mirror vector corresponding to this vector.

This function must return a newly created vector that describes this normal surface in the corresponding "standard" coordinate system (i.e., NSVectorStandard if this subclass is NSVectorQuad, or NSVectorANStandard if this subclass is NSVectorQuadOct, and so on.)

Typically subclasses should implement this function by calling the corresponding static makeMirror() function, which does the real work.

Parameters
triangthe triangulation in which this normal surface lives.
Returns
a newly created vector that describes the same normal surface in a "standard" coordinate system, as described above.

Implements regina::NSVectorMirrored.

◆ octs()

LargeInteger regina::NSVectorQuad::octs ( size_t  tetIndex,
int  octType,
const Triangulation< 3 > *  triang 
) const
inlineoverridevirtualinherited

Returns the number of octagonal discs of the given type in this normal surface.

See NormalSurface::octs() for further details.

Parameters
tetIndexthe index in the triangulation of the tetrahedron in which the requested octagons reside; this should be between 0 and Triangulation<3>::size()-1 inclusive.
octTypethe type of this octagon in the given tetrahedron; this should be 0, 1 or 2.
triangthe triangulation in which this normal surface lives.
Returns
the number of octagonal discs of the given type.

Reimplemented from regina::NSVectorMirrored.

◆ operator+=()

void regina::NSVectorMirrored::operator+= ( const NormalSurfaceVector other)
inlineoverridevirtualinherited

Adds the given vector to this vector.

This behaves correctly in the case where other is this.

The default implementation simply adds the coordinates of the underlying vectors. Subclasses should reimplement this if they carry any additional information that also needs adjusting.

Precondition
This and the given vector represent normal surfaces in the same triangulation, and use the same normal coordinate system.
Parameters
otherthe vector to add to this vector.

Reimplemented from regina::NormalSurfaceVector.

◆ operator[]()

const LargeInteger & regina::NormalSurfaceVector::operator[] ( size_t  index) const
inlineinherited

Returns the given coordinate from the underlying vector.

Parameters
indexthe index of the coordinate to retrieve; this must be between 0 and size()-1 inclusive.
Returns
the coordinate at the given index.

◆ orientedQuads()

LargeInteger regina::NSVectorMirrored::orientedQuads ( size_t  tetIndex,
int  quadType,
const Triangulation< 3 > *  triang,
bool  orientation 
) const
inlineoverridevirtualinherited

Returns the number of oriented quadrilateral discs of the given type in this normal surface.

See NormalSurface::orientedQuads() for further details.

The default implementation of this routine returns zero, which is suitable for coordinate systems that do not support transverse orientation.

Parameters
tetIndexthe index in the triangulation of the tetrahedron in which the requested quadrilaterals reside; this should be between 0 and Triangulation<3>::size()-1 inclusive.
quadTypethe type of this quadrilateral in the given tetrahedron; this should be 0, 1 or 2.
triangthe triangulation in which this normal surface lives.
orientationthe orientation of the normal discs.
Returns
the number of quadrilateral discs of the given type.

Reimplemented from regina::NormalSurfaceVector.

Reimplemented in regina::NSVectorOrientedQuad.

◆ orientedTriangles()

LargeInteger regina::NSVectorMirrored::orientedTriangles ( size_t  tetIndex,
int  vertex,
const Triangulation< 3 > *  triang,
bool  orientation 
) const
inlineoverridevirtualinherited

Returns the number of oriented triangular discs of the given type in this normal surface.

See NormalSurface::orientedTriangles() for further details.

The default implementation of this routine returns zero, which is suitable for coordinate systems that do not support transverse orientation.

Parameters
tetIndexthe index in the triangulation of the tetrahedron in which the requested triangles reside; this should be between 0 and Triangulation<3>::size()-1 inclusive.
vertexthe vertex of the given tetrahedron around which the requested triangles lie; this should be between 0 and 3 inclusive.
triangthe triangulation in which this normal surface lives.
orientationthe orientation of the normal discs.
Returns
the number of triangular discs of the given type.

Reimplemented from regina::NormalSurfaceVector.

◆ quads()

LargeInteger regina::NSVectorMirrored::quads ( size_t  tetIndex,
int  quadType,
const Triangulation< 3 > *  triang 
) const
inlineoverridevirtualinherited

Returns the number of quadrilateral discs of the given type in this normal surface.

See NormalSurface::quads() for further details.

Parameters
tetIndexthe index in the triangulation of the tetrahedron in which the requested quadrilaterals reside; this should be between 0 and Triangulation<3>::size()-1 inclusive.
quadTypethe type of this quadrilateral in the given tetrahedron; this should be 0, 1 or 2.
triangthe triangulation in which this normal surface lives.
Returns
the number of quadrilateral discs of the given type.

Implements regina::NormalSurfaceVector.

Reimplemented in regina::NSVectorOrientedQuad.

◆ scaleDown()

void regina::NSVectorMirrored::scaleDown ( )
inlineoverridevirtualinherited

Scales this vector down by the greatest common divisor of all its elements.

The resulting vector will be the smallest multiple of the original that maintains integral entries, and these entries will have the same signs as the originals.

This routine poses no problem for vectors containing infinite elements; such elements are simply ignored and left at infinity.

The default implementation simply scales down the underlying vector. Subclasses should reimplement this if they carry any additional information that also needs adjusting.

Reimplemented from regina::NormalSurfaceVector.

◆ setElement()

void regina::NSVectorMirrored::setElement ( size_t  index,
const LargeInteger value 
)
inlineoverridevirtualinherited

Sets the given normal coordinate to the given value.

The default implementation simply sets the coordinate in the underlying vector. Subclasses should reimplement this if they carry any additional information that also need adjusting.

Parameters
indexthe index of the coordinate to set; this must be between 0 and size()-1 inclusive.
valuethe new value to assign to the given coordinate.

Reimplemented from regina::NormalSurfaceVector.

◆ size()

size_t regina::NormalSurfaceVector::size ( ) const
inlineinherited

Returns the number of coordinates in the underlying vector.

Returns
the number of coordinates.

◆ triangles()

LargeInteger regina::NSVectorMirrored::triangles ( size_t  tetIndex,
int  vertex,
const Triangulation< 3 > *  triang 
) const
inlineoverridevirtualinherited

Returns the number of triangular discs of the given type in this normal surface.

See NormalSurface::triangles() for further details.

Parameters
tetIndexthe index in the triangulation of the tetrahedron in which the requested triangles reside; this should be between 0 and Triangulation<3>::size()-1 inclusive.
vertexthe vertex of the given tetrahedron around which the requested triangles lie; this should be between 0 and 3 inclusive.
triangthe triangulation in which this normal surface lives.
Returns
the number of triangular discs of the given type.

Implements regina::NormalSurfaceVector.

Member Data Documentation

◆ coords_

Ray regina::NormalSurfaceVector::coords_
protectedinherited

The raw vector of normal coordinates.


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

Copyright © 1999-2021, The Regina development team
This software is released under the GNU General Public License, with some additional permissions; see the source code for details.
For further information, or to submit a bug or other problem, please contact Ben Burton (bab@maths.uq.edu.au).