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regina::NTriangulation Class Reference
[Triangulations]

Stores the triangulation of a 3-manifold along with its various cellular structures and other information. More...

#include <ntriangulation.h>

Inheritance diagram for regina::NTriangulation:

regina::NPacket regina::NFilePropertyReader regina::ShareableObject regina::boost::noncopyable List of all members.

Public Types

typedef NIndexedArray< NTetrahedron *,
HashPointer >::const_iterator 
TetrahedronIterator
 Used to iterate through tetrahedra.

typedef NIndexedArray< NFace *,
HashPointer >::const_iterator 
FaceIterator
 Used to iterate through faces.

typedef NIndexedArray< NEdge *,
HashPointer >::const_iterator 
EdgeIterator
 Used to iterate through edges.

typedef NIndexedArray< NVertex *,
HashPointer >::const_iterator 
VertexIterator
 Used to iterate through vertices.

typedef NIndexedArray< NComponent *,
HashPointer >::const_iterator 
ComponentIterator
 Used to iterate through components.

typedef NIndexedArray< NBoundaryComponent *,
HashPointer >::const_iterator 
BoundaryComponentIterator
 Used to iterate through boundary components.

typedef std::map< std::pair<
unsigned long, unsigned long >,
double > 
TuraevViroSet
 A map from (r, whichRoot) pairs to Turaev-Viro invariants.


Public Member Functions

Constructors and Destructors
 NTriangulation ()
 Default constructor.

 NTriangulation (const NTriangulation &cloneMe)
 Copy constructor.

virtual ~NTriangulation ()
 Destroys this triangulation.

Packet Administration
virtual int getPacketType () const
 Returns the integer ID representing this type of packet.

virtual std::string getPacketTypeName () const
 Returns an English name for this type of packet.

virtual void writePacket (NFile &out) const
 Writes the packet details to the given old-style binary file.

virtual void writeTextShort (std::ostream &out) const
 Writes this object in short text format to the given output stream.

virtual void writeTextLong (std::ostream &out) const
 Writes this object in long text format to the given output stream.

virtual bool dependsOnParent () const
 Determines if this packet depends upon its parent.

virtual void readIndividualProperty (NFile &infile, unsigned propType)
 Reads an individual property from an old-style binary file.

Tetrahedra
unsigned long getNumberOfTetrahedra () const
 Returns the number of tetrahedra in the triangulation.

const NIndexedArray< NTetrahedron *,
HashPointer > & 
getTetrahedra () const
 Returns all tetrahedra in the triangulation.

NTetrahedrongetTetrahedron (unsigned long index)
 Returns the tetrahedron with the given index number in the triangulation.

const NTetrahedrongetTetrahedron (unsigned long index) const
 Returns the tetrahedron with the given index number in the triangulation.

unsigned long getTetrahedronIndex (const NTetrahedron *tet) const
 Returns the index of the given tetrahedron in the triangulation.

void addTetrahedron (NTetrahedron *tet)
 Inserts the given tetrahedron into the triangulation.

NTetrahedronremoveTetrahedron (NTetrahedron *tet)
 Removes the given tetrahedron from the triangulation.

NTetrahedronremoveTetrahedronAt (unsigned long index)
 Removes the tetrahedron with the given index number from the triangulation.

void removeAllTetrahedra ()
 Removes all tetrahedra from the triangulation.

void gluingsHaveChanged ()
 This must be called whenever the gluings of tetrahedra are changed! Clears appropriate properties and performs other necessary tasks.

Skeletal Queries
unsigned long getNumberOfBoundaryComponents () const
 Returns the number of boundary components in this triangulation.

unsigned long getNumberOfComponents () const
 Returns the number of components in this triangulation.

unsigned long getNumberOfVertices () const
 Returns the number of vertices in this triangulation.

unsigned long getNumberOfEdges () const
 Returns the number of edges in this triangulation.

unsigned long getNumberOfFaces () const
 Returns the number of faces in this triangulation.

const NIndexedArray< NComponent *,
HashPointer > & 
getComponents () const
 Returns all components of this triangulation.

const NIndexedArray< NBoundaryComponent *,
HashPointer > & 
getBoundaryComponents () const
 Returns all boundary components of this triangulation.

const NIndexedArray< NVertex *,
HashPointer > & 
getVertices () const
 Returns all vertices of this triangulation.

const NIndexedArray< NEdge *,
HashPointer > & 
getEdges () const
 Returns all edges of this triangulation.

const NIndexedArray< NFace *,
HashPointer > & 
getFaces () const
 Returns all faces of this triangulation.

NComponentgetComponent (unsigned long index) const
 Returns the requested triangulation component.

NBoundaryComponentgetBoundaryComponent (unsigned long index) const
 Returns the requested triangulation boundary component.

NVertexgetVertex (unsigned long index) const
 Returns the requested triangulation vertex.

NEdgegetEdge (unsigned long index) const
 Returns the requested triangulation edge.

NFacegetFace (unsigned long index) const
 Returns the requested triangulation face.

unsigned long getComponentIndex (const NComponent *component) const
 Returns the index of the given component in the triangulation.

unsigned long getBoundaryComponentIndex (const NBoundaryComponent *bc) const
 Returns the index of the given boundary component in the triangulation.

unsigned long getVertexIndex (const NVertex *vertex) const
 Returns the index of the given vertex in the triangulation.

unsigned long getEdgeIndex (const NEdge *edge) const
 Returns the index of the given edge in the triangulation.

unsigned long getFaceIndex (const NFace *face) const
 Returns the index of the given face in the triangulation.

std::auto_ptr< NIsomorphismisIsomorphicTo (const NTriangulation &other) const
 Determines if this triangulation is combinatorially isomorphic to the given triangulation.

std::auto_ptr< NIsomorphismisContainedIn (const NTriangulation &other) const
 Determines if an isomorphic copy of this triangulation is contained within the given triangulation, possibly as a subcomplex of some larger component (or components).

bool hasTwoSphereBoundaryComponents () const
 Determines if this triangulation contains any two-sphere boundary components.

bool hasNegativeIdealBoundaryComponents () const
 Determines if this triangulation contains any ideal boundary components with negative Euler characteristic.

Basic Properties
long getEulerCharacteristic () const
 Returns the Euler characteristic of this triangulation.

bool isValid () const
 Determines if this triangulation is valid.

bool isIdeal () const
 Determines if this triangulation is ideal.

bool isStandard () const
 Determines if this triangulation is standard.

bool hasBoundaryFaces () const
 Determines if this triangulation has any boundary faces.

bool isClosed () const
 Determines if this triangulation is closed.

bool isOrientable () const
 Determines if this triangulation is orientable.

bool isConnected () const
 Determines if this triangulation is connected.

Algebraic Properties
const NGroupPresentationgetFundamentalGroup () const
 Returns the fundamental group of this triangulation.

const NAbelianGroupgetHomologyH1 () const
 Returns the first homology group for this triangulation.

const NAbelianGroupgetHomologyH1Rel () const
 Returns the relative first homology group with respect to the boundary for this triangulation.

const NAbelianGroupgetHomologyH1Bdry () const
 Returns the first homology group of the boundary for this triangulation.

const NAbelianGroupgetHomologyH2 () const
 Returns the second homology group for this triangulation.

unsigned long getHomologyH2Z2 () const
 Returns the second homology group with coefficients in Z_2 for this triangulation.

double turaevViro (unsigned long r, unsigned long whichRoot) const
 Computes the Turaev-Viro state sum invariant of this 3-manifold based upon the given initial data.

const TuraevViroSetallCalculatedTuraevViro () const
 Returns the set of all Turaev-Viro state sum invariants that have already been calculated for this 3-manifold.

Normal Surface Properties
bool isZeroEfficient ()
 Determines if this triangulation is 0-efficient.

bool knowsZeroEfficient () const
 Is it already known whether or not this triangulation is 0-efficient? See isZeroEfficient() for further details.

bool hasSplittingSurface ()
 Determines whether this triangulation has a normal splitting surface.

bool knowsSplittingSurface () const
 Is it already known whether or not this triangulation has a splitting surface? See hasSplittingSurface() for further details.

Skeletal Transformations
void maximalForestInBoundary (stdhash::hash_set< NEdge *, HashPointer > &edgeSet, stdhash::hash_set< NVertex *, HashPointer > &vertexSet) const
 Produces a maximal forest in the 1-skeleton of the triangulation boundary.

void maximalForestInSkeleton (stdhash::hash_set< NEdge *, HashPointer > &edgeSet, bool canJoinBoundaries=true) const
 Produces a maximal forest in the triangulation's 1-skeleton.

void maximalForestInDualSkeleton (stdhash::hash_set< NFace *, HashPointer > &faceSet) const
 Produces a maximal forest in the triangulation's dual 1-skeleton.

bool crushMaximalForest ()
 Attempts to reduce the number of vertices by crushing a maximal forest in the 1-skeleton.

bool intelligentSimplify ()
 Attempts to simplify the triangulation as intelligently as possible without further input.

bool simplifyToLocalMinimum (bool perform=true)
 Uses all known simplification moves to reduce the triangulation monotonically to some local minimum number of tetrahedra.

bool threeTwoMove (NEdge *e, bool check=true, bool perform=true)
 Checks the eligibility of and/or performs a 3-2 move about the given edge.

bool twoThreeMove (NFace *f, bool check=true, bool perform=true)
 Checks the eligibility of and/or performs a 2-3 move about the given face.

bool fourFourMove (NEdge *e, int newAxis, bool check=true, bool perform=true)
 Checks the eligibility of and/or performs a 4-4 move about the given edge.

bool twoZeroMove (NEdge *e, bool check=true, bool perform=true)
 Checks the eligibility of and/or performs a 2-0 move about the given edge of degree 2.

bool twoZeroMove (NVertex *v, bool check=true, bool perform=true)
 Checks the eligibility of and/or performs a 2-0 move about the given vertex of degree 2.

bool twoOneMove (NEdge *e, int edgeEnd, bool check=true, bool perform=true)
 Checks the eligibility of and/or performs a 2-1 move about the given edge.

bool openBook (NFace *f, bool check=true, bool perform=true)
 Checks the eligibility of and/or performs a book opening move about the given face.

bool shellBoundary (NTetrahedron *t, bool check=true, bool perform=true)
 Checks the eligibility of and/or performs a boundary shelling move on the given tetrahedron.

bool collapseEdge (NEdge *e, bool check=true, bool perform=true)
 Checks the eligibility of and/or performs a collapse of an edge in such a way that the topology of the manifold does not change and the number of vertices of the triangulation decreases by one.

Decompositions
unsigned long splitIntoComponents (NPacket *componentParent=0, bool setLabels=true)
 Splits a disconnected triangulation into many smaller triangulations, one for each component.

unsigned long connectedSumDecomposition (NPacket *primeParent=0, bool setLabels=true)
 Splits this triangulation into its connected sum decomposition.

bool isThreeSphere () const
 Determines whether this is a triangulation of a 3-sphere.

bool knowsThreeSphere () const
 Is it already known (or trivial to determine) whether or not this is a triangulation of a 3-sphere? See isThreeSphere() for further details.

NPacketmakeZeroEfficient ()
 Converts this into a 0-efficient triangulation of the same underlying 3-manifold.

Subdivisions and Covers
void makeDoubleCover ()
 Converts this triangulation into its double cover.

bool idealToFinite (bool forceDivision=false)
 Converts an ideal triangulation into a finite triangulation.

void barycentricSubdivision ()
 Does a barycentric subdivision of the triangulation.

Building Triangulations
NTetrahedroninsertLayeredSolidTorus (unsigned long cuts0, unsigned long cuts1)
 Inserts a new layered solid torus into the triangulation.

void insertLayeredLensSpace (unsigned long p, unsigned long q)
 Inserts a new layered lens space L(p,q) into the triangulation.

void insertLayeredLoop (unsigned long length, bool twisted)
 Inserts a layered loop of the given length into this triangulation.

void insertAugTriSolidTorus (long a1, long b1, long a2, long b2, long a3, long b3)
 Inserts an augmented triangular solid torus with the given parameters into this triangulation.

void insertSFSOverSphere (long a1=1, long b1=0, long a2=1, long b2=0, long a3=1, long b3=0)
 Inserts an orientable Seifert fibred space with at most three exceptional fibres over the 2-sphere into this triangulation.

void insertTriangulation (const NTriangulation &source)
 Inserts a copy of the given triangulation into this triangulation.

bool insertRehydration (const std::string &dehydration)
 Inserts the rehydration of the given string into this triangulation.


Static Public Member Functions

NTriangulationenterTextTriangulation (std::istream &in, std::ostream &out)
 Allows the user to interactively enter a triangulation in plain text.

NXMLPacketReadergetXMLReader (NPacket *parent)
 Returns a newly created XML element reader that will read the contents of a single XML packet element.

NTriangulationreadPacket (NFile &in, NPacket *parent)
 Reads a single packet from the specified file and returns a newly created object containing that information.


Static Public Attributes

const int packetType
 Contains the integer ID for this packet.


Protected Member Functions

virtual NPacketinternalClonePacket (NPacket *parent) const
 Makes a newly allocated copy of this packet.

virtual void writeXMLPacketData (std::ostream &out) const
 Writes a chunk of XML containing the data for this packet only.

void cloneFrom (const NTriangulation &from)
 Turns this triangulation into a clone of the given triangulation.


Detailed Description

Stores the triangulation of a 3-manifold along with its various cellular structures and other information.

When the triangulation is deleted, the corresponding tetrahedra, the cellular structure and all other properties will be deallocated.

Faces, edges, vertices and components are always temporary; whenever a change occurs with the triangulation, these will be deleted and a new skeletal structure will be calculated. The same is true of various other triangulation properties.

Whenever the gluings of tetrahedra have been altered, the routine responsible for changing the gluings must call NTriangulation::gluingsHaveChanged() to ensure that relevant properties will be recalculated when necessary. It is not necessary to call this function when adding or removing tetrahedra.

Test:
Test suite contains partial tests.

Todo:
Feature: Is the boundary incompressible?

Feature: Add set of cusps and three corresponding get functions.

Feature (long-term): Am I obviously a handlebody? (Simplify and see if there is nothing left). Am I obviously not a handlebody? (Compare homology with boundary homology).

Feature (long-term): Is the triangulation Haken?

Feature (long-term): What is the Heegaard genus?

Feature (long-term): Have a subcomplex as a child packet of a triangulation. Include routines to crush a subcomplex or to expand a subcomplex to a normal surface.

Feature (long-term): Implement writeTextLong() for skeletal objects.

Feature (long-term): Random triangulation with n tetrahedra.


Member Typedef Documentation

typedef NIndexedArray<NBoundaryComponent*, HashPointer>::const_iterator regina::NTriangulation::BoundaryComponentIterator
 

Used to iterate through boundary components.

typedef NIndexedArray<NComponent*, HashPointer>::const_iterator regina::NTriangulation::ComponentIterator
 

Used to iterate through components.

typedef NIndexedArray<NEdge*, HashPointer>::const_iterator regina::NTriangulation::EdgeIterator
 

Used to iterate through edges.

typedef NIndexedArray<NFace*, HashPointer>::const_iterator regina::NTriangulation::FaceIterator
 

Used to iterate through faces.

typedef NIndexedArray<NTetrahedron*, HashPointer>::const_iterator regina::NTriangulation::TetrahedronIterator
 

Used to iterate through tetrahedra.

typedef std::map<std::pair<unsigned long, unsigned long>, double> regina::NTriangulation::TuraevViroSet
 

A map from (r, whichRoot) pairs to Turaev-Viro invariants.

typedef NIndexedArray<NVertex*, HashPointer>::const_iterator regina::NTriangulation::VertexIterator
 

Used to iterate through vertices.


Constructor & Destructor Documentation

regina::NTriangulation::NTriangulation  )  [inline]
 

Default constructor.

Creates an empty triangulation.

regina::NTriangulation::NTriangulation const NTriangulation cloneMe  )  [inline]
 

Copy constructor.

Creates a new triangulation identical to the given triangulation. The packet tree structure and packet label are not copied.

Parameters:
cloneMe the triangulation to clone.

regina::NTriangulation::~NTriangulation  )  [inline, virtual]
 

Destroys this triangulation.

The contained tetrahedra, the cellular structure and all other properties will also be deallocated.


Member Function Documentation

void regina::NTriangulation::addTetrahedron NTetrahedron tet  )  [inline]
 

Inserts the given tetrahedron into the triangulation.

No face gluings anywhere will be examined or altered.

The new tetrahedron will be assigned a higher index in the triangulation than all tetrahedra already present.

There is no need to call gluingsHaveChanged() after calling this function.

Python:
Since this triangulation takes ownership of the given tetrahedron, the python object containing the given tetrahedron becomes a null object and should no longer be used.
Parameters:
tet the tetrahedron to insert.

const NTriangulation::TuraevViroSet & regina::NTriangulation::allCalculatedTuraevViro  )  const [inline]
 

Returns the set of all Turaev-Viro state sum invariants that have already been calculated for this 3-manifold.

Turaev-Viro invariants are described by an (r, whichRoot) pair as described in the turaevViro() notes. The set returned by this routine maps (r, whichRoot) pairs to the corresponding invariant values.

Each time turaevViro() is called, the result will be stored in this set (as well as being returned to the user). This set will be emptied whenever the triangulation is modified.

Python:
Not present.
Returns:
the set of all Turaev-Viro invariants that have already been calculated.
See also:
turaevViro

void regina::NTriangulation::barycentricSubdivision  ) 
 

Does a barycentric subdivision of the triangulation.

Each tetrahedron is divided into 24 tetrahedra by placing an extra vertex at the centroid of each tetrahedron, the centroid of each face and the midpoint of each edge.

Author:
David Letscher

void regina::NTriangulation::cloneFrom const NTriangulation from  )  [protected]
 

Turns this triangulation into a clone of the given triangulation.

The tree structure and label of this triangulation are not touched.

Parameters:
from the triangulation from which this triangulation will be cloned.

bool regina::NTriangulation::collapseEdge NEdge e,
bool  check = true,
bool  perform = true
 

Checks the eligibility of and/or performs a collapse of an edge in such a way that the topology of the manifold does not change and the number of vertices of the triangulation decreases by one.

If the routine is asked to both check and perform, the move will only be performed if the check shows it is legal.

Note that after performing this move, all skeletal objects (faces, components, etc.) will be invalid.

Precondition:
If the move is being performed and no check is being run, it must be known in advance that the move is legal.

The skeleton has been calculated. Skeleton calculation can be forced by querying the skeleton, such as calling getNumberOfVertices().

Warning:
This routine should not be used until the eligibility checks are corrected; see the bug details below.

Todo:
Bug (urgent): The restrictions on allowing this move to take place are currently wrong. Many valid cases are ruled out (as acknowledged in the original commit of the code), but certain invalid cases are also allowed which will almost certainly crash the program.
Parameters:
e the edge to collapse.
check true if we are to check whether the move is allowed (defaults to true).
perform true if we are to perform the move (defaults to true).
Returns:
If check is true, the function returns true if and only if the given edge may be collapsed without changing the topology of the manifold. If check is false, the function simply returns true.
Author:
David Letscher

unsigned long regina::NTriangulation::connectedSumDecomposition NPacket primeParent = 0,
bool  setLabels = true
 

Splits this triangulation into its connected sum decomposition.

The individual prime 3-manifold triangulations that make up this decomposition will be inserted as children of the given parent packet. The original triangulation will be left unchanged.

Note that this routine is currently only available for closed orientable triangulations; see the list of preconditions for full details. The 0-efficiency prime decomposition algorithm of Jaco and Rubinstein is used.

If the given parent packet is 0, the new prime summand triangulations will be inserted as children of this triangulation.

This routine can optionally assign unique (and sensible) packet labels to each of the new prime summand triangulations. Note however that uniqueness testing may be slow, so this assignment of labels should be disabled if the summand triangulations are only temporary objects used as part of a larger routine.

If this is a triangulation of a 3-sphere, no prime summand triangulations will be created at all.

Warning:
The algorithms used in this routine rely on normal surface theory and so can be very slow for larger triangulations. For 3-sphere testing, see the routine isThreeSphere() which uses faster methods where possible.
Precondition:
This triangulation is valid, closed, orientable and connected.
Parameters:
primeParent the packet beneath which the new prime summand triangulations will be inserted, or 0 if they should be inserted directly beneath this triangulation.
setLabels true if the new prime summand triangulations should be assigned unique packet labels, or false if they should be left without labels at all.
Returns:
the number of prime summands created, 0 if this triangulation is a 3-sphere or 0 if this triangulation does not meet the preconditions described above.

bool regina::NTriangulation::crushMaximalForest  ) 
 

Attempts to reduce the number of vertices by crushing a maximal forest in the 1-skeleton.

Todo:
Bug (urgent): This algorithm needs to be changed from the current incorrect algorithm to Dave's algorithm that avoids crisis by using 2-3 moves.
Returns:
true if and only if the triangulation was changed.

bool regina::NTriangulation::dependsOnParent  )  const [inline, virtual]
 

Determines if this packet depends upon its parent.

This is true if the parent cannot be altered without invalidating or otherwise upsetting this packet.

Returns:
true if and only if this packet depends on its parent.

Implements regina::NPacket.

NTriangulation* regina::NTriangulation::enterTextTriangulation std::istream &  in,
std::ostream &  out
[static]
 

Allows the user to interactively enter a triangulation in plain text.

Prompts will be sent to the given output stream and information will be read from the given input stream.

Python:
This routine is a member of class Engine. It takes no parameters; in and out are always assumed to be standard input and standard output respectively.
Parameters:
in the input stream from which text will be read.
out the output stream to which prompts will be written.
Returns:
the triangulation entered in by the user.

bool regina::NTriangulation::fourFourMove NEdge e,
int  newAxis,
bool  check = true,
bool  perform = true
 

Checks the eligibility of and/or performs a 4-4 move about the given edge.

This involves replacing the four tetrahedra joined at that edge with four tetrahedra joined along a different edge. Consider the octahedron made up of the four original tetrahedra; this has three internal axes. The initial four tetrahedra meet along the given edge which forms one of these axes; the new tetrahedra will meet along a different axis. This move can be done iff the edge is non-boundary and the four tetrahedra are distinct.

If the routine is asked to both check and perform, the move will only be performed if the check shows it is legal.

Note that after performing this move, all skeletal objects (faces, components, etc.) will be invalid.

Precondition:
If the move is being performed and no check is being run, it must be known in advance that the move is legal.

The skeleton has been calculated. Skeleton calculation can be forced by querying the skeleton, such as calling getNumberOfVertices().

Parameters:
e the edge about which to perform the move.
newAxis Specifies which axis of the octahedron the new tetrahedra should meet along; this should be 0 or 1. Consider the four original tetrahedra in the order described by NEdge::getEmbeddings(); call these tetrahedra 0, 1, 2 and 3. If newAxis is 0, the new axis will separate tetrahedra 0 and 1 from 2 and 3. If newAxis is 1, the new axis will separate tetrahedra 1 and 2 from 3 and 0.
check true if we are to check whether the move is allowed (defaults to true).
perform true if we are to perform the move (defaults to true).
Returns:
If check is true, the function returns true if and only if the requested move may be performed without changing the topology of the manifold. If check is false, the function simply returns true.

NBoundaryComponent * regina::NTriangulation::getBoundaryComponent unsigned long  index  )  const [inline]
 

Returns the requested triangulation boundary component.

Bear in mind that each time the triangulation changes, the boundary components will be deleted and replaced with new ones. Thus this object should be considered temporary only.

Parameters:
index the index of the desired boundary component, ranging from 0 to getNumberOfBoundaryComponents()-1 inclusive.
Returns:
the requested boundary component.

unsigned long regina::NTriangulation::getBoundaryComponentIndex const NBoundaryComponent bc  )  const [inline]
 

Returns the index of the given boundary component in the triangulation.

Precondition:
The given boundary component belongs to this triangulation.
Parameters:
bc specifies which boundary component to find in the triangulation.
Returns:
the index of the specified boundary component, where 0 is the first boundary component, 1 is the second and so on.

const NIndexedArray< NBoundaryComponent *, HashPointer > & regina::NTriangulation::getBoundaryComponents  )  const [inline]
 

Returns all boundary components of this triangulation.

Note that each ideal vertex forms its own boundary component.

Bear in mind that each time the triangulation changes, the boundary components will be deleted and replaced with new ones. Thus the objects contained in this list should be considered temporary only.

This reference to the list however will remain valid and up-to-date for as long as the triangulation exists.

Python:
This routine returns a python list.
Returns:
the list of all boundary components.

NComponent * regina::NTriangulation::getComponent unsigned long  index  )  const [inline]
 

Returns the requested triangulation component.

Bear in mind that each time the triangulation changes, the components will be deleted and replaced with new ones. Thus this object should be considered temporary only.

Parameters:
index the index of the desired component, ranging from 0 to getNumberOfComponents()-1 inclusive.
Returns:
the requested component.

unsigned long regina::NTriangulation::getComponentIndex const NComponent component  )  const [inline]
 

Returns the index of the given component in the triangulation.

Precondition:
The given component belongs to this triangulation.
Parameters:
component specifies which component to find in the triangulation.
Returns:
the index of the specified component, where 0 is the first component, 1 is the second and so on.

const NIndexedArray< NComponent *, HashPointer > & regina::NTriangulation::getComponents  )  const [inline]
 

Returns all components of this triangulation.

Bear in mind that each time the triangulation changes, the components will be deleted and replaced with new ones. Thus the objects contained in this list should be considered temporary only.

This reference to the list however will remain valid and up-to-date for as long as the triangulation exists.

Python:
This routine returns a python list.
Returns:
the list of all components.

NEdge * regina::NTriangulation::getEdge unsigned long  index  )  const [inline]
 

Returns the requested triangulation edge.

Bear in mind that each time the triangulation changes, the edges will be deleted and replaced with new ones. Thus this object should be considered temporary only.

Parameters:
index the index of the desired edge, ranging from 0 to getNumberOfEdges()-1 inclusive.
Returns:
the requested edge.

unsigned long regina::NTriangulation::getEdgeIndex const NEdge edge  )  const [inline]
 

Returns the index of the given edge in the triangulation.

Precondition:
The given edge belongs to this triangulation.
Parameters:
edge specifies which edge to find in the triangulation.
Returns:
the index of the specified edge, where 0 is the first edge, 1 is the second and so on.

const NIndexedArray< NEdge *, HashPointer > & regina::NTriangulation::getEdges  )  const [inline]
 

Returns all edges of this triangulation.

Bear in mind that each time the triangulation changes, the edges will be deleted and replaced with new ones. Thus the objects contained in this list should be considered temporary only.

This reference to the list however will remain valid and up-to-date for as long as the triangulation exists.

Python:
This routine returns a python list.
Returns:
the list of all edges.

long regina::NTriangulation::getEulerCharacteristic  )  const [inline]
 

Returns the Euler characteristic of this triangulation.

This will be evaluated strictly as V-E+F-T. Thus if the manifold contains cusps, the Euler characteristic will almost certainly not be the same as the corresponding 3-manifold with the cusps truncated.

Returns:
the Euler characteristic.

NFace * regina::NTriangulation::getFace unsigned long  index  )  const [inline]
 

Returns the requested triangulation face.

Bear in mind that each time the triangulation changes, the faces will be deleted and replaced with new ones. Thus this object should be considered temporary only.

Parameters:
index the index of the desired face, ranging from 0 to getNumberOfFaces()-1 inclusive.
Returns:
the requested face.

unsigned long regina::NTriangulation::getFaceIndex const NFace face  )  const [inline]
 

Returns the index of the given face in the triangulation.

Precondition:
The given face belongs to this triangulation.
Parameters:
face specifies which face to find in the triangulation.
Returns:
the index of the specified face, where 0 is the first face, 1 is the second and so on.

const NIndexedArray< NFace *, HashPointer > & regina::NTriangulation::getFaces  )  const [inline]
 

Returns all faces of this triangulation.

Bear in mind that each time the triangulation changes, the faces will be deleted and replaced with new ones. Thus the objects contained in this list should be considered temporary only.

This reference to the list however will remain valid and up-to-date for as long as the triangulation exists.

Python:
This routine returns a python list.
Returns:
the list of all faces.

const NGroupPresentation& regina::NTriangulation::getFundamentalGroup  )  const
 

Returns the fundamental group of this triangulation.

If this triangulation contains any ideal or non-standard vertices, the fundamental group will be calculated as if each such vertex had been truncated.

If this triangulation contains any invalid edges, the calculations will be performed without any truncation of the corresponding projective plane cusp. Thus if a barycentric subdivision is performed on the triangulation, the result of getFundamentalGroup() will change.

Bear in mind that each time the triangulation changes, the fundamental group will be deleted. Thus the group reference returned should not be kept for later use. Instead, getFundamentalGroup() should be called again; this will be instantaneous if the group has already been calculated.

Note that this triangulation is not required to be valid (see isValid()).

Precondition:
This triangulation has at most one component.
Returns:
the fundamental group.

const NAbelianGroup& regina::NTriangulation::getHomologyH1  )  const
 

Returns the first homology group for this triangulation.

If this triangulation contains any ideal or non-standard vertices, the homology group will be calculated as if each such vertex had been truncated.

If this triangulation contains any invalid edges, the calculations will be performed without any truncation of the corresponding projective plane cusp. Thus if a barycentric subdivision is performed on the triangulation, the result of getHomologyH1() will change.

Bear in mind that each time the triangulation changes, the homology groups will be deleted. Thus the group reference returned should not be kept for later use. Instead, getHomologyH1() should be called again; this will be instantaneous if the group has already been calculated.

Note that this triangulation is not required to be valid (see isValid()).

Returns:
the first homology group.

const NAbelianGroup& regina::NTriangulation::getHomologyH1Bdry  )  const
 

Returns the first homology group of the boundary for this triangulation.

Note that ideal vertices are considered part of the boundary.

Bear in mind that each time the triangulation changes, the homology groups will be deleted. Thus the group reference returned should not be kept for later use. Instead, getHomologyH1Bdry() should be called again; this will be instantaneous if the group has already been calculated.

This routine is fairly fast, since it deduces the homology of each boundary component through knowing what kind of surface it is.

Precondition:
This triangulation is valid.
Returns:
the first homology group of the boundary.

const NAbelianGroup& regina::NTriangulation::getHomologyH1Rel  )  const
 

Returns the relative first homology group with respect to the boundary for this triangulation.

Note that ideal vertices are considered part of the boundary.

Bear in mind that each time the triangulation changes, the homology groups will be deleted. Thus the group reference returned should not be kept for later use. Instead, getHomologyH1Rel() should be called again; this will be instantaneous if the group has already been calculated.

Precondition:
This triangulation is valid.
Returns:
the relative first homology group with respect to the boundary.

const NAbelianGroup& regina::NTriangulation::getHomologyH2  )  const
 

Returns the second homology group for this triangulation.

If this triangulation contains any ideal vertices, the homology group will be calculated as if each such vertex had been truncated. The algorithm used calculates various first homology groups and uses homology and cohomology theorems to deduce the second homology group.

Bear in mind that each time the triangulation changes, the homology groups will be deleted. Thus the group reference returned should not be kept for later use. Instead, getHomologyH2() should be called again; this will be instantaneous if the group has already been calculated.

Precondition:
This triangulation is valid.
Returns:
the second homology group.

unsigned long regina::NTriangulation::getHomologyH2Z2  )  const [inline]
 

Returns the second homology group with coefficients in Z_2 for this triangulation.

If this triangulation contains any ideal vertices, the homology group will be calculated as if each such vertex had been truncated. The algorithm used calculates the relative first homology group with respect to the boundary and uses homology and cohomology theorems to deduce the second homology group.

This group will simply be the direct sum of several copies of Z_2, so the number of Z_2 terms is returned.

Precondition:
This triangulation is valid.
Returns:
the number of Z_2 terms in the second homology group with coefficients in Z_2.

unsigned long regina::NTriangulation::getNumberOfBoundaryComponents  )  const [inline]
 

Returns the number of boundary components in this triangulation.

Note that each ideal vertex forms its own boundary component.

Returns:
the number of boundary components.

unsigned long regina::NTriangulation::getNumberOfComponents  )  const [inline]
 

Returns the number of components in this triangulation.

Returns:
the number of components.

unsigned long regina::NTriangulation::getNumberOfEdges  )  const [inline]
 

Returns the number of edges in this triangulation.

Returns:
the number of edges.

unsigned long regina::NTriangulation::getNumberOfFaces  )  const [inline]
 

Returns the number of faces in this triangulation.

Returns:
the number of faces.

unsigned long regina::NTriangulation::getNumberOfTetrahedra  )  const [inline]
 

Returns the number of tetrahedra in the triangulation.

Returns:
the number of tetrahedra.

unsigned long regina::NTriangulation::getNumberOfVertices  )  const [inline]
 

Returns the number of vertices in this triangulation.

Returns:
the number of vertices.

virtual int regina::NTriangulation::getPacketType  )  const [virtual]
 

Returns the integer ID representing this type of packet.

This is the same for all packets of this class.

Returns:
the packet type ID.

Implements regina::NPacket.

virtual std::string regina::NTriangulation::getPacketTypeName  )  const [virtual]
 

Returns an English name for this type of packet.

An example is NTriangulation. This is the same for all packets of this class.

Returns:
the packet type name.

Implements regina::NPacket.

const NIndexedArray< NTetrahedron *, HashPointer > & regina::NTriangulation::getTetrahedra  )  const [inline]
 

Returns all tetrahedra in the triangulation.

The reference returned will remain valid for as long as the triangulation exists, always reflecting the tetrahedra currently in the triangulation.

Python:
This routine returns a python list.
Returns:
the list of all tetrahedra.

const NTetrahedron * regina::NTriangulation::getTetrahedron unsigned long  index  )  const [inline]
 

Returns the tetrahedron with the given index number in the triangulation.

Note that tetrahedron indexing may change when a tetrahedron is added or removed from the triangulation.

This routine will ensure the skeleton is calculated, since other skeleton objects can be accessed from NTetrahedron.

Parameters:
index specifies which tetrahedron to return; this value should be between 0 and getNumberOfTetrahedra()-1 inclusive.
Returns:
the indexth tetrahedron in the triangulation.

NTetrahedron * regina::NTriangulation::getTetrahedron unsigned long  index  )  [inline]
 

Returns the tetrahedron with the given index number in the triangulation.

Note that tetrahedron indexing may change when a tetrahedron is added or removed from the triangulation.

This routine will ensure the skeleton is calculated, since other skeleton objects can be accessed from NTetrahedron.

Parameters:
index specifies which tetrahedron to return; this value should be between 0 and getNumberOfTetrahedra()-1 inclusive.
Returns:
the indexth tetrahedron in the triangulation.

unsigned long regina::NTriangulation::getTetrahedronIndex const NTetrahedron tet  )  const [inline]
 

Returns the index of the given tetrahedron in the triangulation.

Note that tetrahedron indexing may change when a tetrahedron is added or removed from the triangulation.

Parameters:
tet specifies which tetrahedron to find in the triangulation.
Returns:
the index of the specified tetrahedron, where 0 is the first tetrahedron, 1 is the second and so on. If the tetrahedron is not contained in the triangulation, a negative number is returned.

NVertex * regina::NTriangulation::getVertex unsigned long  index  )  const [inline]
 

Returns the requested triangulation vertex.

Bear in mind that each time the triangulation changes, the vertices will be deleted and replaced with new ones. Thus this object should be considered temporary only.

Parameters:
index the index of the desired vertex, ranging from 0 to getNumberOfVertices()-1 inclusive.
Returns:
the requested vertex.

unsigned long regina::NTriangulation::getVertexIndex const NVertex vertex  )  const [inline]
 

Returns the index of the given vertex in the triangulation.

Precondition:
The given vertex belongs to this triangulation.
Parameters:
vertex specifies which vertex to find in the triangulation.
Returns:
the index of the specified vertex, where 0 is the first vertex, 1 is the second and so on.

const NIndexedArray< NVertex *, HashPointer > & regina::NTriangulation::getVertices  )  const [inline]
 

Returns all vertices of this triangulation.

Bear in mind that each time the triangulation changes, the vertices will be deleted and replaced with new ones. Thus the objects contained in this list should be considered temporary only.

This reference to the list however will remain valid and up-to-date for as long as the triangulation exists.

Python:
This routine returns a python list.
Returns:
the list of all vertices.

NXMLPacketReader* regina::NTriangulation::getXMLReader NPacket parent  )  [static]
 

Returns a newly created XML element reader that will read the contents of a single XML packet element.

You may assume that the packet to be read is of the same type as the class in which you are implementing this routine.

The XML element reader should read exactly what writeXMLPacketData() writes, and vice versa.

parent represents the packet which will become the new packet's parent in the tree structure, and may be assumed to have already been read from the file. This information is for reference only, and does not need to be used. The XML element reader can either insert or not insert the new packet beneath parent in the tree structure as it pleases. Note however that parent will be 0 if the new packet is to become a tree matriarch.

This routine is not actually provided for NPacket itself, but must be declared and implemented for every packet subclass that will be instantiated.

Python:
Not present.
Parameters:
parent the packet which will become the new packet's parent in the tree structure, or 0 if the new packet is to be tree matriarch.
Returns:
the newly created XML element reader.

Reimplemented from regina::NPacket.

void regina::NTriangulation::gluingsHaveChanged  )  [inline]
 

This must be called whenever the gluings of tetrahedra are changed! Clears appropriate properties and performs other necessary tasks.

The responsibility of calling gluingsHaveChanged() falls upon the routine that alters the gluings (such as a component of a triangulation editor, or so on).

bool regina::NTriangulation::hasBoundaryFaces  )  const [inline]
 

Determines if this triangulation has any boundary faces.

Returns:
true if and only if there are boundary faces.

bool regina::NTriangulation::hasNegativeIdealBoundaryComponents  )  const [inline]
 

Determines if this triangulation contains any ideal boundary components with negative Euler characteristic.

Returns:
true if and only if there is at least one such boundary component.

bool regina::NTriangulation::hasSplittingSurface  ) 
 

Determines whether this triangulation has a normal splitting surface.

See NNormalSurface::isSplitting() for details regarding normal splitting surfaces.

Precondition:
This triangulation is connected. If the triangulation is not connected, this routine will still return a result but that result will be unreliable.
Returns:
true if and only if this triangulation has a normal splitting surface.

bool regina::NTriangulation::hasTwoSphereBoundaryComponents  )  const [inline]
 

Determines if this triangulation contains any two-sphere boundary components.

Returns:
true if and only if there is at least one two-sphere boundary component.

bool regina::NTriangulation::idealToFinite bool  forceDivision = false  ) 
 

Converts an ideal triangulation into a finite triangulation.

All ideal or non-standard vertices are truncated and thus converted into real boundary components made from unglued faces of tetrahedra.

Warning:
Currently, this routine subdivides all tetrahedra as if all vertices (not just some) were ideal. This may lead to more tetrahedra than are necessary.

Currently, the presence of an invalid edge will force the triangulation to be subdivided regardless of the value of parameter forceDivision. The final triangulation will still have the projective plane cusp caused by the invalid edge.

Todo:
Optimise (long-term): Have this routine only use as many tetrahedra as are necessary, leaving finite vertices alone.
Parameters:
forceDivision specifies what to do if the triangulation has no ideal or non-standard vertices. If true, the triangulation will be subdivided anyway, as if all vertices were ideal. If false (the default), the triangulation will be left alone.
Returns:
true if and only if the triangulation was changed.
Author:
David Letscher

void regina::NTriangulation::insertAugTriSolidTorus long  a1,
long  b1,
long  a2,
long  b2,
long  a3,
long  b3
 

Inserts an augmented triangular solid torus with the given parameters into this triangulation.

Almost all augmented triangular solid tori represent Seifert fibred spaces with three or fewer exceptional fibres. Augmented triangular solid tori are described in more detail in the NAugTriSolidTorus class notes.

The resulting Seifert fibred space will be SFS((a1,b1) (a2,b2) (a3,b3) (1,1)), where the parameters a1, ..., b3 are passed as arguments to this routine. The three layered solid tori that are attached to the central triangular solid torus will be LST(|a1|, |b1|, |-a1-b1|), ..., LST(|a3|, |b3|, |-a3-b3|).

The new tetrahedra will be inserted at the end of the list of tetrahedra in the triangulation.

Precondition:
gcd(a1, b1) = 1.

gcd(a2, b2) = 1.

gcd(a3, b3) = 1.

Parameters:
a1 a parameter describing the first layered solid torus in the augmented triangular solid torus; this may be either positive or negative.
b1 a parameter describing the first layered solid torus in the augmented triangular solid torus; this may be either positive or negative.
a2 a parameter describing the second layered solid torus in the augmented triangular solid torus; this may be either positive or negative.
b2 a parameter describing the second layered solid torus in the augmented triangular solid torus; this may be either positive or negative.
a3 a parameter describing the third layered solid torus in the augmented triangular solid torus; this may be either positive or negative.
b3 a parameter describing the third layered solid torus in the augmented triangular solid torus; this may be either positive or negative.

void regina::NTriangulation::insertLayeredLensSpace unsigned long  p,
unsigned long  q
 

Inserts a new layered lens space L(p,q) into the triangulation.

The lens space will be created by gluing together two layered solid tori in a way that uses the fewest possible tetrahedra.

The new tetrahedra will be inserted at the end of the list of tetrahedra in the triangulation.

Precondition:
p > q >= 0 unless (p,q) = (0,1);

gcd(p, q) = 1.

Parameters:
p a parameter of the desired lens space.
q a parameter of the desired lens space.
See also:
NLayeredLensSpace

void regina::NTriangulation::insertLayeredLoop unsigned long  length,
bool  twisted
 

Inserts a layered loop of the given length into this triangulation.

Layered loops are described in more detail in the NLayeredLoop class notes.

The new tetrahedra will be inserted at the end of the list of tetrahedra in the triangulation.

Parameters:
length the length of the new layered loop; this must be strictly positive.
twisted true if the new layered loop should be twisted, or false if it should be untwisted.
See also:
NLayeredLoop

NTetrahedron* regina::NTriangulation::insertLayeredSolidTorus unsigned long  cuts0,
unsigned long  cuts1
 

Inserts a new layered solid torus into the triangulation.

The meridinal disc of the layered solid torus will intersect the three edges of the boundary torus in cuts0, cuts1 and (cuts0 + cuts1) points respectively.

The boundary torus will always consist of faces 012 and 013 of the tetrahedron containing this boundary torus (this tetrahedron will be returned). In face 012, edges 12, 02 and 01 will meet the meridinal disc cuts0, cuts1 and (cuts0 + cuts1) times respectively. The only exceptions are if these three intersection numbers are (1,1,2) or (0,1,1), in which case edges 12, 02 and 01 will meet the meridinal disc (1, 2 and 1) or (1, 1 and 0) times respectively.

The new tetrahedra will be inserted at the end of the list of tetrahedra in the triangulation.

Precondition:
0 <= cuts0 <= cuts1;

cuts1 is non-zero;

gcd(cuts0, cuts1) = 1.

Parameters:
cuts0 the smallest of the three desired intersection numbers.
cuts1 the second smallest of the three desired intersection numbers.
Returns:
the tetrahedron containing the boundary torus.
See also:
NLayeredSolidTorus

bool regina::NTriangulation::insertRehydration const std::string &  dehydration  ) 
 

Inserts the rehydration of the given string into this triangulation.

The given string will be rehydrated into a proper triangulation. The new tetrahedra will be inserted into this triangulation in the order in which they appear in the rehydrated triangulation, and the numbering of their vertices (0-3) will not change.

For a full description of the dehydrated triangulation format, see A Census of Cusped Hyperbolic 3-Manifolds, Callahan, Hildebrand and Weeks, Mathematics of Computation 68/225, 1999.

Parameters:
dehydration a dehydrated representation of the triangulation to insert. Case is irrelevant; all letters will be treated as if they were lower case.
Returns:
true if the insertion was successful, or false if the given string could not be rehydrated.

void regina::NTriangulation::insertSFSOverSphere long  a1 = 1,
long  b1 = 0,
long  a2 = 1,
long  b2 = 0,
long  a3 = 1,
long  b3 = 0
 

Inserts an orientable Seifert fibred space with at most three exceptional fibres over the 2-sphere into this triangulation.

The inserted Seifert fibred space will be SFS((a1,b1) (a2,b2) (a3,b3) (1,1)), where the parameters a1, ..., b3 are passed as arguments to this routine.

The three pairs of parameters (a,b) do not need to be normalised, i.e., the parameters can be positive or negative and b may lie outside the range [0..a). There is no separate twisting parameter; each additional twist can be incorporated into the existing parameters by replacing some pair (a,b) with the pair (a,a+b). For Seifert fibred spaces with less than three exceptional fibres, some or all of the parameter pairs may be (1,k) or even (1,0).

The new tetrahedra will be inserted at the end of the list of tetrahedra in the triangulation.

Precondition:
None of a1, a2 or a3 are 0.

gcd(a1, b1) = 1.

gcd(a2, b2) = 1.

gcd(a3, b3) = 1.

Parameters:
a1 a parameter describing the first exceptional fibre.
b1 a parameter describing the first exceptional fibre.
a2 a parameter describing the second exceptional fibre.
b2 a parameter describing the second exceptional fibre.
a3 a parameter describing the third exceptional fibre.
b3 a parameter describing the third exceptional fibre.

void regina::NTriangulation::insertTriangulation const NTriangulation source  ) 
 

Inserts a copy of the given triangulation into this triangulation.

The new tetrahedra will be inserted into this triangulation in the order in which they appear in the given triangulation, and the numbering of their vertices (0-3) will not change. They will be given the same descriptions as appear in the given triangulation.

Parameters:
source the triangulation whose copy will be inserted.

bool regina::NTriangulation::intelligentSimplify  ) 
 

Attempts to simplify the triangulation as intelligently as possible without further input.

Currently this routine merely uses simplifyToLocalMinimum() in combination with random 4-4 moves.

Warning:
The specific behaviour of this routine is very likely to change between releases.

Todo:
Optimise (urgent): Make this faster and more effective. Include book opening moves and random 2-3 moves to get out of wells. Unglue faces with three boundary edges and record the corresponding change in topology. Minimise the amount of skeletal/homological calculation.
Returns:
true if and only if the triangulation was changed.

NPacket * regina::NTriangulation::internalClonePacket NPacket parent  )  const [inline, protected, virtual]
 

Makes a newly allocated copy of this packet.

This routine should not insert the new packet into the tree structure, clone the packet's associated tags or give the packet a label. It should also not clone any descendants of this packet.

You may assume that the new packet will eventually be inserted into the tree beneath either the same parent as this packet or a clone of that parent.

Parameters:
parent the parent beneath which the new packet will eventually be inserted.
Returns:
the newly allocated packet.

Implements regina::NPacket.

bool regina::NTriangulation::isClosed  )  const [inline]
 

Determines if this triangulation is closed.

This is the case if and only if it has no boundary. Note that ideal triangulations are not closed.

Returns:
true if and only if this triangulation is closed.

bool regina::NTriangulation::isConnected  )  const [inline]
 

Determines if this triangulation is connected.

Returns:
true if and only if this triangulation is connected.

std::auto_ptr<NIsomorphism> regina::NTriangulation::isContainedIn const NTriangulation other  )  const
 

Determines if an isomorphic copy of this triangulation is contained within the given triangulation, possibly as a subcomplex of some larger component (or components).

Specifically, this routine determines if there is a boundary incomplete combinatorial isomorphism from this triangulation to other. Boundary incomplete isomorphisms are described in detail in the NIsomorphism class notes.

In particular, note that boundary faces of this triangulation need not correspond to boundary faces of other, and that other can contain more tetrahedra than this triangulation.

If a boundary incomplete isomorphism is found, the details of this isomorphism are returned. The isomorphism is newly constructed, and so to assist with memory management is returned as a std::auto_ptr. Thus, to test whether an isomorphism exists without having to explicitly deal with the isomorphism itself, you can call if (isContainedIn(other).get()) and the newly created isomorphism (if it exists) will be automatically destroyed.

Parameters:
other the triangulation in which to search for an isomorphic copy of this triangulation.
Returns:
details of the isomorphism if such a copy is found, or a null pointer otherwise.

bool regina::NTriangulation::isIdeal  )  const [inline]
 

Determines if this triangulation is ideal.

This is the case if and only if one of the vertex links is closed and not a 2-sphere. Note that the triangulation is not required to be valid.

Returns:
true if and only if this triangulation is ideal.

std::auto_ptr<NIsomorphism> regina::NTriangulation::isIsomorphicTo const NTriangulation other  )  const
 

Determines if this triangulation is combinatorially isomorphic to the given triangulation.

Specifically, this routine determines if there is a one-to-one and onto boundary complete combinatorial isomorphism from this triangulation to other. Boundary complete isomorphisms are described in detail in the NIsomorphism class notes.

In particular, note that this triangulation and other must contain the same number of tetrahedra for such an isomorphism to exist.

Todo:
Optimise: Improve the complexity by choosing a tetrahedron mapping from each component and following gluings to determine the others.
If a boundary complete isomorphism is found, the details of this isomorphism are returned. The isomorphism is newly constructed, and so to assist with memory management is returned as a std::auto_ptr. Thus, to test whether an isomorphism exists without having to explicitly deal with the isomorphism itself, you can call if (isIsomorphicTo(other).get()) and the newly created isomorphism (if it exists) will be automatically destroyed.

Parameters:
other the triangulation to compare with this one.
Returns:
details of the isomorphism if the two triangulations are combinatorially isomorphic, or a null pointer otherwise.

bool regina::NTriangulation::isOrientable  )  const [inline]
 

Determines if this triangulation is orientable.

Returns:
true if and only if this triangulation is orientable.

bool regina::NTriangulation::isStandard  )  const [inline]
 

Determines if this triangulation is standard.

This is the case if and only if every vertex is standard. See NVertex::isStandard() for further details.

Returns:
true if and only if this triangulation is standard.

bool regina::NTriangulation::isThreeSphere  )  const
 

Determines whether this is a triangulation of a 3-sphere.

This routine relies upon a combination of Rubinstein's 3-sphere recognition algorithm and Jaco and Rubinstein's 0-efficiency prime decomposition algorithm.

Warning:
The algorithms used in this routine rely on normal surface theory and so can be very slow for larger triangulations (although faster tests are used where possible). The routine knowsThreeSphere() can be called to see if this property is already known or if it happens to be very fast to calculate for this triangulation.
Returns:
true if and only if this is a 3-sphere triangulation.

bool regina::NTriangulation::isValid  )  const [inline]
 

Determines if this triangulation is valid.

A triangulation is valid unless there is some vertex whose link has boundary but is not a disc (i.e., a vertex for which NVertex::getLink() returns NVertex::NON_STANDARD_BDRY), or unless there is some edge glued to itself in reverse (i.e., an edge for which NEdge::isValid() returns false).

Returns:
true if and only if this triangulation is valid.

bool regina::NTriangulation::isZeroEfficient  ) 
 

Determines if this triangulation is 0-efficient.

A triangulation is 0-efficient if its only normal spheres and discs are vertex linking, and if it has no 2-sphere boundary components.

Returns:
true if and only if this triangulation is 0-efficient.

bool regina::NTriangulation::knowsSplittingSurface  )  const [inline]
 

Is it already known whether or not this triangulation has a splitting surface? See hasSplittingSurface() for further details.

If this property is already known, future calls to hasSplittingSurface() will be very fast (simply returning the precalculated value).

Returns:
true if and only if this property is already known.

bool regina::NTriangulation::knowsThreeSphere  )  const
 

Is it already known (or trivial to determine) whether or not this is a triangulation of a 3-sphere? See isThreeSphere() for further details.

If this property is indeed already known, future calls to isThreeSphere() will be very fast (simply returning the precalculated value).

If this property is not already known, this routine will nevertheless run some very fast preliminary tests to see if the answer is obviously no. If so, it will store false as the precalculated value for isThreeSphere() and this routine will return true.

Otherwise a call to isThreeSphere() may potentially require more significant work, and so this routine will return false.

Returns:
true if and only if this property is already known or trivial to calculate.

bool regina::NTriangulation::knowsZeroEfficient  )  const [inline]
 

Is it already known whether or not this triangulation is 0-efficient? See isZeroEfficient() for further details.

If this property is already known, future calls to isZeroEfficient() will be very fast (simply returning the precalculated value).

Returns:
true if and only if this property is already known.

void regina::NTriangulation::makeDoubleCover  ) 
 

Converts this triangulation into its double cover.

Each orientable component will be duplicated, and each non-orientable component will be converted into its orientable double cover.

NPacket* regina::NTriangulation::makeZeroEfficient  ) 
 

Converts this into a 0-efficient triangulation of the same underlying 3-manifold.

A triangulation is 0-efficient if its only normal spheres and discs are vertex linking, and if it has no 2-sphere boundary components.

Note that this routine is currently only available for closed orientable triangulations; see the list of preconditions for details. The 0-efficiency algorithm of Jaco and Rubinstein is used.

If the underlying 3-manifold is prime, it can always be made 0-efficient (with the exception of the special cases RP3 and S2xS1 as noted below). In this case the original triangulation will be modified directly and 0 will be returned.

If the underyling 3-manifold is RP3 or S2xS1, it cannot be made 0-efficient; in this case the original triangulation will be reduced to a two-tetrahedron minimal triangulation and 0 will again be returned.

If the underlying 3-manifold is not prime, it cannot be made 0-efficient. In this case the original triangulation will remain unchanged and a new connected sum decomposition will be returned. This will be presented as a newly allocated container packet with one child triangulation for each prime summand.

Warning:
The algorithms used in this routine rely on normal surface theory and so can be very slow for larger triangulations.
Precondition:
This triangulation is valid, closed, orientable and connected.
Returns:
0 if the underlying 3-manifold is prime (in which case the original triangulation was modified directly), or a newly allocated connected sum decomposition if the underlying 3-manifold is composite (in which case the original triangulation was not changed).

void regina::NTriangulation::maximalForestInBoundary stdhash::hash_set< NEdge *, HashPointer > &  edgeSet,
stdhash::hash_set< NVertex *, HashPointer > &  vertexSet
const
 

Produces a maximal forest in the 1-skeleton of the triangulation boundary.

Both given sets will be emptied and the edges and vertices of the maximal forest will be placed into them. A vertex that forms its own boundary component (such as an ideal vertex) will still be placed in vertexSet.

Note that the edge and vertex pointers returned will become invalid once the triangulation has changed.

Python:
Not present.
Parameters:
edgeSet the set to be emptied and into which the edges of the maximal forest will be placed.
vertexSet the set to be emptied and into which the vertices of the maximal forest will be placed.

void regina::NTriangulation::maximalForestInDualSkeleton stdhash::hash_set< NFace *, HashPointer > &  faceSet  )  const
 

Produces a maximal forest in the triangulation's dual 1-skeleton.

The given set will be emptied and will have the faces corresponding to the edges of the maximal forest in the dual 1-skeleton placed into it.

Note that the face pointers returned will become invalid once the triangulation has changed.

Python:
Not present.
Parameters:
faceSet the set to be emptied and into which the faces representing the maximal forest will be placed.

void regina::NTriangulation::maximalForestInSkeleton stdhash::hash_set< NEdge *, HashPointer > &  edgeSet,
bool  canJoinBoundaries = true
const
 

Produces a maximal forest in the triangulation's 1-skeleton.

The given set will be emptied and will have the edges of the maximal forest placed into it. It can be specified whether or not different boundary components may be joined by the maximal forest.

An edge leading to an ideal vertex is still a candidate for inclusion in the maximal forest. For the purposes of this algorithm, any ideal vertex will be treated as any other vertex (and will still be considered part of its own boundary component).

Note that the edge pointers returned will become invalid once the triangulation has changed.

Python:
Not present.
Parameters:
edgeSet the set to be emptied and into which the edges of the maximal forest will be placed.
canJoinBoundaries true if and only if different boundary components are allowed to be joined by the maximal forest.

bool regina::NTriangulation::openBook NFace f,
bool  check = true,
bool  perform = true
 

Checks the eligibility of and/or performs a book opening move about the given face.

This involves taking a face meeting the boundary along two edges and ungluing it to create two new boundary faces and thus expose the tetrahedra it initially joined, allowing for potential boundary shelling moves. This move can be done only if the face meets the boundary in precisely two edges (and thus also joins two tetrahedra) and if the vertex between these two edges is a standard boundary vertex (its link is a disc).

If the routine is asked to both check and perform, the move will only be performed if the check shows it is legal.

Note that after performing this move, all skeletal objects (faces, components, etc.) will be invalid.

Precondition:
If the move is being performed and no check is being run, it must be known in advance that the move is legal.

The skeleton has been calculated. Skeleton calculation can be forced by querying the skeleton, such as calling getNumberOfVertices().

Parameters:
f the face about which to perform the move.
check true if we are to check whether the move is allowed (defaults to true).
perform true if we are to perform the move (defaults to true).
Returns:
If check is true, the function returns true if and only if the requested move may be performed without changing the topology of the manifold. If check is false, the function simply returns true.

virtual void regina::NTriangulation::readIndividualProperty NFile infile,
unsigned  propType
[virtual]
 

Reads an individual property from an old-style binary file.

The property type and bookmarking details should not read; merely the contents of the property that are written to file between NFile::writePropertyHeader() and NFile::writePropertyFooter(). See the NFile::writePropertyHeader() notes for details.

The property type of the property to be read will be passed in propType. If the property type is unrecognised, this routine should simply do nothing and return. If the property type is recognised, this routine should read the property and process it accordingly (e.g., store it in whatever data object is currently being read).

Parameters:
infile the file from which to read the property. This should be open for reading and at the position immediately after writePropertyHeader() would have been called during the corresponding write operation.
propType the property type of the property about to be read.

Implements regina::NFilePropertyReader.

NTriangulation* regina::NTriangulation::readPacket NFile in,
NPacket parent
[static]
 

Reads a single packet from the specified file and returns a newly created object containing that information.

You may assume that the packet to be read is of the same type as the class in which you are implementing this routine. The newly created object must also be of this type.

For instance, NTriangulation::readPacket() may assume that the packet is of type NTriangulation, and must return a pointer to a newly created NTriangulation. Deallocation of the newly created packet is the responsibility of whoever calls this routine.

The packet type and label may be assumed to have already been read from the file, and should not be reread. The readPacket() routine should read exactly what writePacket() writes, and vice versa.

parent represents the packet which will become the new packet's parent in the tree structure, and may be assumed to have already been read from the file. This information is for reference only, and does not need to be used. This routine can either insert or not insert the new packet beneath parent in the tree structure as it pleases. Note however that parent will be 0 if the new packet is to become a tree matriarch.

This routine is not actually provided for NPacket itself, but must be declared and implemented for every packet subclass that will be instantiated. Within each such subclass the function must be declared to return a pointer to an object of that subclass. For instance, NTriangulation::readPacket() must be declared to return an NTriangulation*, not simply an NPacket*.

New packet types should make this routine simply return 0 since this file format is now obsolete, and older calculation engines will not understand newer packet types anyway.

Deprecated:
For the preferred way to read packets from file, see getXMLReader() and class NXMLPacketReader instead.
Precondition:
The given file is open for reading and all above conditions have been satisfied.
Python:
Not present.
Parameters:
in the file from which to read the packet.
parent the packet which will become the new packet's parent in the tree structure, or 0 if the new packet is to be tree matriarch.
Returns:
the packet read from file, or 0 if an error occurred.

Reimplemented from regina::NPacket.

void regina::NTriangulation::removeAllTetrahedra  )  [inline]
 

Removes all tetrahedra from the triangulation.

All tetrahedra will be deallocated.

There is no need to call gluingsHaveChanged() after calling this function.

NTetrahedron * regina::NTriangulation::removeTetrahedron NTetrahedron tet  )  [inline]
 

Removes the given tetrahedron from the triangulation.

All faces glued to this tetrahedron will be unglued. The tetrahedron will not be deallocated.

There is no need to call gluingsHaveChanged() after calling this function.

Precondition:
The given tetrahedron exists in the triangulation.
Parameters:
tet the tetrahedron to remove.
Returns:
the removed tetrahedron.

NTetrahedron * regina::NTriangulation::removeTetrahedronAt unsigned long  index  )  [inline]
 

Removes the tetrahedron with the given index number from the triangulation.

Note that tetrahedron indexing may change when a tetrahedron is added or removed from the triangulation.

All faces glued to this tetrahedron will be unglued. The tetrahedron will not be deallocated.

There is no need to call gluingsHaveChanged() after calling this function.

Parameters:
index specifies which tetrahedron to remove; this should be between 0 and getNumberOfTetrahedra()-1 inclusive.
Returns:
the removed tetrahedron.

bool regina::NTriangulation::shellBoundary NTetrahedron t,
bool  check = true,
bool  perform = true
 

Checks the eligibility of and/or performs a boundary shelling move on the given tetrahedron.

This involves simply popping off a tetrahedron that touches the boundary. This can be done only if precisely 1, 2 or 3 faces of the tetrahedron lie in the boundary. Furthermore, if 1 face lies in the boundary, the opposite vertex may not lie in the boundary. If 2 faces lie in the boundary, the remaining edge may not lie in the boundary and the remaining two faces of the tetrahedron may not be glued together.

If the routine is asked to both check and perform, the move will only be performed if the check shows it is legal.

Note that after performing this move, all skeletal objects (faces, components, etc.) will be invalid.

Precondition:
If the move is being performed and no check is being run, it must be known in advance that the move is legal.

The skeleton has been calculated. Skeleton calculation can be forced by querying the skeleton, such as calling getNumberOfVertices().

Parameters:
t the tetrahedron upon which to perform the move.
check true if we are to check whether the move is allowed (defaults to true).
perform true if we are to perform the move (defaults to true).
Returns:
If check is true, the function returns true if and only if the requested move may be performed without changing the topology of the manifold. If check is false, the function simply returns true.

bool regina::NTriangulation::simplifyToLocalMinimum bool  perform = true  ) 
 

Uses all known simplification moves to reduce the triangulation monotonically to some local minimum number of tetrahedra.

Note that this will probably not give a globally minimal triangulation; see intelligentSimplify() for further assistance in achieving this goal.

The moves used include 3-2, 2-0 (edge and vertex), 2-1 and boundary shelling moves.

Note that book opening moves (which do not reduce the number of tetrahedra) are no longer used in this routine, in contrast with earlier releases of Regina.

Warning:
The specific behaviour of this routine is very likely to change between releases.

Todo:
Bug (urgent): This routine currently does not crush a maximal forest!
Parameters:
perform true if we are to perform the simplifications, or false if we are only to investigate whether simplifications are possible (defaults to true).
Returns:
if perform is true, this routine returns true if and only if the triangulation was changed to reduce the number of tetrahedra; if perform is false, this routine returns true if and only if it determines that it is capable of performing such a change.

unsigned long regina::NTriangulation::splitIntoComponents NPacket componentParent = 0,
bool  setLabels = true
 

Splits a disconnected triangulation into many smaller triangulations, one for each component.

The new component triangulations will be inserted as children of the given parent packet. The original triangulation will be left unchanged.

If the given parent packet is 0, the new component triangulations will be inserted as children of this triangulation.

This routine can optionally assign unique (and sensible) packet labels to each of the new component triangulations. Note however that uniqueness testing may be slow, so this assignment of labels should be disabled if the component triangulations are only temporary objects used as part of a larger routine.

Parameters:
componentParent the packet beneath which the new component triangulations will be inserted, or 0 if they should be inserted directly beneath this triangulation.
setLabels true if the new component triangulations should be assigned unique packet labels, or false if they should be left without labels at all.
Returns:
the number of new component triangulations constructed.

bool regina::NTriangulation::threeTwoMove NEdge e,
bool  check = true,
bool  perform = true
 

Checks the eligibility of and/or performs a 3-2 move about the given edge.

This involves replacing the three tetrahedra joined at that edge with two tetrahedra joined by a face. This can be done iff the edge is non-boundary and the three tetrahedra are distinct.

If the routine is asked to both check and perform, the move will only be performed if the check shows it is legal.

Note that after performing this move, all skeletal objects (faces, components, etc.) will be invalid.

Precondition:
If the move is being performed and no check is being run, it must be known in advance that the move is legal.

The skeleton has been calculated. Skeleton calculation can be forced by querying the skeleton, such as calling getNumberOfVertices().

Parameters:
e the edge about which to perform the move.
check true if we are to check whether the move is allowed (defaults to true).
perform true if we are to perform the move (defaults to true).
Returns:
If check is true, the function returns true if and only if the requested move may be performed without changing the topology of the manifold. If check is false, the function simply returns true.

double regina::NTriangulation::turaevViro unsigned long  r,
unsigned long  whichRoot
const
 

Computes the Turaev-Viro state sum invariant of this 3-manifold based upon the given initial data.

The initial data is as described in the paper of Turaev and Viro, "State sum invariants of 3-manifolds and quantum 6j-symbols", Topology, vol. 31, no. 4, 1992, pp 865-902.

In particular, Section 7 describes the initial data as determined by an integer r >=3 and a root of unity q0 of degree 2r for which q0^2 is a primitive root of unity of degree r.

I believe that these invariants, although computed in the complex field, are all reals. However, I have not yet found a firm assertion of this and so in the meantime this routine simply returns the real portion and sends a warning to standard error if the imaginary portion is non-zero.

Precondition:
This triangulation is valid, closed and non-empty.
Parameters:
r the integer r as described above; this must be at least 3.
whichRoot determines q0 to be the root of unity e^(2i * Pi * whichRoot / 2r); this argument must be strictly between 0 and 2r and must have no common factors with r.
Returns:
the requested Turaev-Viro invariant.
See also:
allCalculatedTuraevViro

bool regina::NTriangulation::twoOneMove NEdge e,
int  edgeEnd,
bool  check = true,
bool  perform = true
 

Checks the eligibility of and/or performs a 2-1 move about the given edge.

This involves taking an edge meeting only one tetrahedron just once and merging that tetrahedron with one of the tetrahedra joining it.

This can be done assuming the following conditions. The edge must be non-boundary. The two vertices that are its endpoints cannot both be boundary. The two remaining faces of the tetrahedron may not be joined. Furthermore, consider the two edges of the second tetrahedron (to be merged) that run from the (identical) vertices of the original tetrahedron not touching e to the vertex of the second tetrahedron not touching the original tetrahedron. These edges must be distinct and may not both be in the boundary. Finally (which should follow from the previous conditions), the two faces joining these two edges to the vertex of e that is common to both tetrahedra should be distinct. Phew. Code documentation could really do with diagrams!

If the routine is asked to both check and perform, the move will only be performed if the check shows it is legal.

Note that after performing this move, all skeletal objects (faces, components, etc.) will be invalid.

Precondition:
If the move is being performed and no check is being run, it must be known in advance that the move is legal.

The skeleton has been calculated. Skeleton calculation can be forced by querying the skeleton, such as calling getNumberOfVertices().

Parameters:
e the edge about which to perform the move.
edgeEnd the end of the edge opposite that at which the second tetrahedron (to be merged) is joined. The end is 0 or 1, corresponding to the labelling (0,1) of the vertices of the edge as described in NEdgeEmbedding::getVertices().
check true if we are to check whether the move is allowed (defaults to true).
perform true if we are to perform the move (defaults to true).
Returns:
If check is true, the function returns true if and only if the requested move may be performed without changing the topology of the manifold. If check is false, the function simply returns true.

bool regina::NTriangulation::twoThreeMove NFace f,
bool  check = true,
bool  perform = true
 

Checks the eligibility of and/or performs a 2-3 move about the given face.

This involves replacing the two tetrahedra joined at that face with three tetrahedra joined by an edge. This can be done iff the two tetrahedra are distinct.

If the routine is asked to both check and perform, the move will only be performed if the check shows it is legal.

Note that after performing this move, all skeletal objects (faces, components, etc.) will be invalid.

Precondition:
If the move is being performed and no check is being run, it must be known in advance that the move is legal.

The skeleton has been calculated. Skeleton calculation can be forced by querying the skeleton, such as calling getNumberOfVertices().

Parameters:
f the face about which to perform the move.
check true if we are to check whether the move is allowed (defaults to true).
perform true if we are to perform the move (defaults to true).
Returns:
If check is true, the function returns true if and only if the requested move may be performed without changing the topology of the manifold. If check is false, the function simply returns true.

bool regina::NTriangulation::twoZeroMove NVertex v,
bool  check = true,
bool  perform = true
 

Checks the eligibility of and/or performs a 2-0 move about the given vertex of degree 2.

This involves taking the two tetrahedra joined at that vertex and squashing them flat. This can be done only if the vertex is non-boundary, the two tetrahedra are distinct, the faces opposite v in each tetrahedron are distinct and not both boundary, and the two tetrahedra meet each other on all three faces touching the vertex (as opposed to meeting each other on one face and being glued to themselves along the other two).

If the routine is asked to both check and perform, the move will only be performed if the check shows it is legal.

Note that after performing this move, all skeletal objects (faces, components, etc.) will be invalid.

Precondition:
If the move is being performed and no check is being run, it must be known in advance that the move is legal.

The skeleton has been calculated. Skeleton calculation can be forced by querying the skeleton, such as calling getNumberOfVertices().

Parameters:
v the vertex about which to perform the move.
check true if we are to check whether the move is allowed (defaults to true).
perform true if we are to perform the move (defaults to true).
Returns:
If check is true, the function returns true if and only if the requested move may be performed without changing the topology of the manifold. If check is false, the function simply returns true.

bool regina::NTriangulation::twoZeroMove NEdge e,
bool  check = true,
bool  perform = true
 

Checks the eligibility of and/or performs a 2-0 move about the given edge of degree 2.

This involves taking the two tetrahedra joined at that edge and squashing them flat. This can be done only if the edge is non-boundary, the two tetrahedra are distinct and the edges opposite e in each tetrahedron are distinct and not both boundary. Furthermore, if faces f1 and f2 of one tetrahedron are to be flattened onto faces g1 and g2 of the other respectively, we must have (a) f1 and g1 distinct, (b) f2 and g2 distinct, (c) not both f1=g2 and g1=f2, (d) not both f1=f2 and g1=g2 and (e) not two of the faces boundary with the other two identified.

If the routine is asked to both check and perform, the move will only be performed if the check shows it is legal.

Note that after performing this move, all skeletal objects (faces, components, etc.) will be invalid.

Precondition:
If the move is being performed and no check is being run, it must be known in advance that the move is legal.

The skeleton has been calculated. Skeleton calculation can be forced by querying the skeleton, such as calling getNumberOfVertices().

Parameters:
e the edge about which to perform the move.
check true if we are to check whether the move is allowed (defaults to true).
perform true if we are to perform the move (defaults to true).
Returns:
If check is true, the function returns true if and only if the requested move may be performed without changing the topology of the manifold. If check is false, the function simply returns true.

virtual void regina::NTriangulation::writePacket NFile out  )  const [virtual]
 

Writes the packet details to the given old-style binary file.

You may assume that the packet type and label have already been written. Only the actual data stored in the packet need be written.

The default implementation for this routine does nothing; new packet types should not implement this routine since this file format is now obsolete, and older calculation engines will simply skip unknown packet types when reading from binary files.

Deprecated:
For the preferred way to write packets to file, see writeXMLFile() and writeXMLPacketData() instead.
Precondition:
The given file is open for writing and satisfies the assumptions listed above.
Python:
Not present.
Parameters:
out the file to be written to.

Reimplemented from regina::NPacket.

virtual void regina::NTriangulation::writeTextLong std::ostream &  out  )  const [virtual]
 

Writes this object in long text format to the given output stream.

The output should provided the user with all the information they could want. The output should end with a newline.

The default implementation of this routine merely calls writeTextShort() and adds a newline.

Python:
The parameter out does not exist; standard output will be used.
Parameters:
out the output stream to which to write.

Reimplemented from regina::ShareableObject.

void regina::NTriangulation::writeTextShort std::ostream &  out  )  const [inline, virtual]
 

Writes this object in short text format to the given output stream.

The output should fit on a single line and no newline should be written.

Python:
The parameter out does not exist; standard output will be used.
Parameters:
out the output stream to which to write.

Implements regina::ShareableObject.

virtual void regina::NTriangulation::writeXMLPacketData std::ostream &  out  )  const [protected, virtual]
 

Writes a chunk of XML containing the data for this packet only.

You may assume that the packet opening tag (including the packet type and label) has already been written, and that all child packets followed by the corresponding packet closing tag will be written immediately after this routine is called. This routine need only write the internal data stored in this specific packet.

Parameters:
out the output stream to which the XML should be written.

Implements regina::NPacket.


Member Data Documentation

const int regina::NTriangulation::packetType [static]
 

Contains the integer ID for this packet.

Each distinct packet type must have a unique ID, and this should be a positive integer. See packetregistry.h for further requirements regarding ID selection.

This member is not actually provided for NPacket itself, but must be declared for every packet subclass that will be instantiated. A value need not be assigned; packetregistry.h will take care of this task when you register the packet.

Reimplemented from regina::NPacket.


The documentation for this class was generated from the following file:
Copyright © 1999-2004, Ben Burton
This software is released under the GNU General Public License.
For further information, or to submit a bug or other problem, please contact Ben Burton (bab@debian.org).