Regina Calculation Engine
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regina::NTriangulation Class Reference

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

#include <triangulation/ntriangulation.h>

Inheritance diagram for regina::NTriangulation:
regina::NPacket regina::NGenericTriangulation< 3 > regina::ShareableObject regina::DimTraits< dim > regina::boost::noncopyable

Public Types

typedef std::vector
< NTetrahedron * >
::const_iterator 
TetrahedronIterator
 Used to iterate through tetrahedra. More...
 
typedef std::vector< NTriangle * >
::const_iterator 
TriangleIterator
 Used to iterate through triangles. More...
 
typedef std::vector< NTriangle * >
::const_iterator 
FaceIterator
 A deprecated alias for TriangleIterator. More...
 
typedef std::vector< NEdge * >
::const_iterator 
EdgeIterator
 Used to iterate through edges. More...
 
typedef std::vector< NVertex * >
::const_iterator 
VertexIterator
 Used to iterate through vertices. More...
 
typedef std::vector
< NComponent * >
::const_iterator 
ComponentIterator
 Used to iterate through components. More...
 
typedef std::vector
< NBoundaryComponent * >
::const_iterator 
BoundaryComponentIterator
 Used to iterate through boundary components. More...
 
typedef std::map< std::pair
< unsigned long, unsigned long >
, double > 
TuraevViroSet
 A map from (r, whichRoot) pairs to Turaev-Viro invariants. More...
 
- Public Types inherited from regina::NPacket
typedef ChangeEventSpan ChangeEventBlock
 A deprecated typedef for ChangeEventSpan. More...
 

Public Member Functions

Constructors and Destructors
 NTriangulation ()
 Default constructor. More...
 
 NTriangulation (const NTriangulation &cloneMe)
 Copy constructor. More...
 
 NTriangulation (const std::string &description)
 "Magic" constructor that tries to find some way to interpret the given string as a triangulation. More...
 
virtual ~NTriangulation ()
 Destroys this triangulation. More...
 
Packet Administration
virtual void writeTextShort (std::ostream &out) const
 Writes this object in short text format to the given output stream. More...
 
virtual void writeTextLong (std::ostream &out) const
 Writes this object in long text format to the given output stream. More...
 
virtual bool dependsOnParent () const
 Determines if this packet depends upon its parent. More...
 
Tetrahedra
unsigned long getNumberOfTetrahedra () const
 Returns the number of tetrahedra in the triangulation. More...
 
unsigned long getNumberOfSimplices () const
 A dimension-agnostic alias for getNumberOfTetrahedra(). More...
 
const std::vector
< NTetrahedron * > & 
getTetrahedra () const
 Returns all tetrahedra in the triangulation. More...
 
const std::vector
< NTetrahedron * > & 
getSimplices () const
 A dimension-agnostic alias for getTetrahedra(). More...
 
NTetrahedrongetTetrahedron (unsigned long index)
 Returns the tetrahedron with the given index number in the triangulation. More...
 
NTetrahedrongetSimplex (unsigned long index)
 A dimension-agnostic alias for getTetrahedron(). More...
 
const NTetrahedrongetTetrahedron (unsigned long index) const
 Returns the tetrahedron with the given index number in the triangulation. More...
 
const NTetrahedrongetSimplex (unsigned long index) const
 A dimension-agnostic alias for getTetrahedron(). More...
 
long tetrahedronIndex (const NTetrahedron *tet) const
 Returns the index of the given tetrahedron in the triangulation. More...
 
long simplexIndex (const NTetrahedron *tet) const
 A dimension-agnostic alias for tetrahedronIndex(). More...
 
NTetrahedronnewTetrahedron ()
 Creates a new tetrahedron and adds it to this triangulation. More...
 
NTetrahedronnewSimplex ()
 A dimension-agnostic alias for newTetrahedron(). More...
 
NTetrahedronnewTetrahedron (const std::string &desc)
 Creates a new tetrahedron with the given description and adds it to this triangulation. More...
 
NTetrahedronnewSimplex (const std::string &desc)
 A dimension-agnostic alias for newTetrahedron(). More...
 
void addTetrahedron (NTetrahedron *tet)
 Inserts the given tetrahedron into the triangulation. More...
 
void removeTetrahedron (NTetrahedron *tet)
 Removes the given tetrahedron from the triangulation. More...
 
void removeSimplex (NTetrahedron *tet)
 A dimension-agnostic alias for removeTetrahedron(). More...
 
void removeTetrahedronAt (unsigned long index)
 Removes the tetrahedron with the given index number from the triangulation. More...
 
void removeSimplexAt (unsigned long index)
 A dimension-agnostic alias for removeTetrahedronAt(). More...
 
void removeAllTetrahedra ()
 Removes all tetrahedra from the triangulation. More...
 
void removeAllSimplices ()
 A dimension-agnostic alias for removeAllTetrahedra(). More...
 
void swapContents (NTriangulation &other)
 Swaps the contents of this and the given triangulation. More...
 
void moveContentsTo (NTriangulation &dest)
 Moves the contents of this triangulation into the given destination triangulation, without destroying any pre-existing contents. More...
 
void gluingsHaveChanged ()
 This routine now does nothing, and should not be used. More...
 
Skeletal Queries
unsigned long getNumberOfBoundaryComponents () const
 Returns the number of boundary components in this triangulation. More...
 
unsigned long getNumberOfComponents () const
 Returns the number of components in this triangulation. More...
 
unsigned long getNumberOfVertices () const
 Returns the number of vertices in this triangulation. More...
 
unsigned long getNumberOfEdges () const
 Returns the number of edges in this triangulation. More...
 
unsigned long getNumberOfTriangles () const
 Returns the number of triangular faces in this triangulation. More...
 
unsigned long getNumberOfFaces () const
 A deprecated alias for getNumberOfTriangles(). More...
 
template<int dim>
unsigned long getNumberOfFaces () const
 Returns the number of faces of the given dimension in this triangulation. More...
 
const std::vector< NComponent * > & getComponents () const
 Returns all components of this triangulation. More...
 
const std::vector
< NBoundaryComponent * > & 
getBoundaryComponents () const
 Returns all boundary components of this triangulation. More...
 
const std::vector< NVertex * > & getVertices () const
 Returns all vertices of this triangulation. More...
 
const std::vector< NEdge * > & getEdges () const
 Returns all edges of this triangulation. More...
 
const std::vector< NTriangle * > & getTriangles () const
 Returns all triangular faces of this triangulation. More...
 
const std::vector< NTriangle * > & getFaces () const
 A deprecated alias for getTriangles(). More...
 
NComponentgetComponent (unsigned long index) const
 Returns the requested triangulation component. More...
 
NBoundaryComponentgetBoundaryComponent (unsigned long index) const
 Returns the requested triangulation boundary component. More...
 
NVertexgetVertex (unsigned long index) const
 Returns the requested vertex in this triangulation. More...
 
NEdgegetEdge (unsigned long index) const
 Returns the requested edge in this triangulation. More...
 
NTrianglegetTriangle (unsigned long index) const
 Returns the requested triangular face in this triangulation. More...
 
NTrianglegetFace (unsigned long index) const
 A deprecated alias for getTriangle(). More...
 
long componentIndex (const NComponent *component) const
 Returns the index of the given component in the triangulation. More...
 
long boundaryComponentIndex (const NBoundaryComponent *bc) const
 Returns the index of the given boundary component in the triangulation. More...
 
long vertexIndex (const NVertex *vertex) const
 Returns the index of the given vertex in the triangulation. More...
 
long edgeIndex (const NEdge *edge) const
 Returns the index of the given edge in the triangulation. More...
 
long triangleIndex (const NTriangle *triangle) const
 Returns the index of the given triangle in the triangulation. More...
 
long faceIndex (const NTriangle *triangle) const
 A deprecated alias for triangleIndex(). More...
 
bool hasTwoSphereBoundaryComponents () const
 Determines if this triangulation contains any two-sphere boundary components. More...
 
bool hasNegativeIdealBoundaryComponents () const
 Determines if this triangulation contains any ideal boundary components with negative Euler characteristic. More...
 
Isomorphism Testing
std::auto_ptr< NIsomorphismisIsomorphicTo (const NTriangulation &other) const
 Determines if this triangulation is combinatorially isomorphic to the given triangulation. More...
 
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). More...
 
unsigned long findAllSubcomplexesIn (const NTriangulation &other, std::list< NIsomorphism * > &results) const
 Finds all ways in which an isomorphic copy of this triangulation is contained within the given triangulation, possibly as a subcomplex of some larger component (or components). More...
 
Basic Properties
long getEulerCharTri () const
 Returns the Euler characteristic of this triangulation. More...
 
long getEulerCharManifold () const
 Returns the Euler characteristic of the corresponding compact 3-manifold. More...
 
long getEulerCharacteristic () const
 A deprecated alias for getEulerCharTri(). More...
 
bool isValid () const
 Determines if this triangulation is valid. More...
 
bool isIdeal () const
 Determines if this triangulation is ideal. More...
 
bool isStandard () const
 Determines if this triangulation is standard. More...
 
bool hasBoundaryTriangles () const
 Determines if this triangulation has any boundary triangles. More...
 
bool hasBoundaryFaces () const
 A deprecated alias for hasBoundaryTriangles(). More...
 
bool isClosed () const
 Determines if this triangulation is closed. More...
 
bool isOrientable () const
 Determines if this triangulation is orientable. More...
 
bool isOriented () const
 Determines if this triangulation is oriented; that is, if tetrahedron vertices are labelled in a way that preserves orientation across adjacent tetrahedron faces. More...
 
bool isOrdered () const
 Determines if this triangulation is ordered; that is, if tetrahedron vertices are labelled so that all gluing permutations are order-preserving on the tetrahedron faces. More...
 
bool isConnected () const
 Determines if this triangulation is connected. More...
 
Algebraic Properties
const NGroupPresentationgetFundamentalGroup () const
 Returns the fundamental group of this triangulation. More...
 
void simplifiedFundamentalGroup (NGroupPresentation *newGroup)
 Notifies the triangulation that you have simplified the presentation of its fundamental group. More...
 
const NAbelianGroupgetHomologyH1 () const
 Returns the first homology group for this triangulation. More...
 
const NAbelianGroupgetHomologyH1Rel () const
 Returns the relative first homology group with respect to the boundary for this triangulation. More...
 
const NAbelianGroupgetHomologyH1Bdry () const
 Returns the first homology group of the boundary for this triangulation. More...
 
const NAbelianGroupgetHomologyH2 () const
 Returns the second homology group for this triangulation. More...
 
unsigned long getHomologyH2Z2 () const
 Returns the second homology group with coefficients in Z_2 for this triangulation. More...
 
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. More...
 
const TuraevViroSetallCalculatedTuraevViro () const
 Returns the set of all Turaev-Viro state sum invariants that have already been calculated for this 3-manifold. More...
 
Normal Surface Properties
bool isZeroEfficient ()
 Determines if this triangulation is 0-efficient. More...
 
bool knowsZeroEfficient () const
 Is it already known whether or not this triangulation is 0-efficient? See isZeroEfficient() for further details. More...
 
bool hasSplittingSurface ()
 Determines whether this triangulation has a normal splitting surface. More...
 
bool knowsSplittingSurface () const
 Is it already known whether or not this triangulation has a splitting surface? See hasSplittingSurface() for further details. More...
 
NNormalSurfacehasNonTrivialSphereOrDisc ()
 Searches for a non-vertex-linking normal sphere or disc within this triangulation. More...
 
NNormalSurfacehasOctagonalAlmostNormalSphere ()
 Searches for an octagonal almost normal 2-sphere within this triangulation. More...
 
Skeletal Transformations
void maximalForestInBoundary (std::set< NEdge * > &edgeSet, std::set< NVertex * > &vertexSet) const
 Produces a maximal forest in the 1-skeleton of the triangulation boundary. More...
 
void maximalForestInSkeleton (std::set< NEdge * > &edgeSet, bool canJoinBoundaries=true) const
 Produces a maximal forest in the triangulation's 1-skeleton. More...
 
void maximalForestInDualSkeleton (std::set< NTriangle * > &triangleSet) const
 Produces a maximal forest in the triangulation's dual 1-skeleton. More...
 
bool intelligentSimplify ()
 Attempts to simplify the triangulation as intelligently as possible without further input. More...
 
bool simplifyToLocalMinimum (bool perform=true)
 Uses all known simplification moves to reduce the triangulation monotonically to some local minimum number of tetrahedra. More...
 
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. More...
 
bool twoThreeMove (NTriangle *t, bool check=true, bool perform=true)
 Checks the eligibility of and/or performs a 2-3 move about the given triangle. More...
 
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. More...
 
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. More...
 
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. More...
 
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. More...
 
bool openBook (NTriangle *t, bool check=true, bool perform=true)
 Checks the eligibility of and/or performs a book opening move about the given triangle. More...
 
bool closeBook (NEdge *e, bool check=true, bool perform=true)
 Checks the eligibility of and/or performs a book closing move about the given boundary edge. More...
 
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. More...
 
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. More...
 
void reorderTetrahedraBFS (bool reverse=false)
 Reorders the tetrahedra of this triangulation using a breadth-first search, so that small-numbered tetrahedra are adjacent to other small-numbered tetrahedra. More...
 
void orient ()
 Relabels tetrahedron vertices in this triangulation so that all tetrahedra are oriented consistently, if possible. More...
 
bool order (bool forceOriented=false)
 Relabels tetrahedron vertices in this triangulation to give an ordered triangulation, if possible. More...
 
Decompositions
unsigned long splitIntoComponents (NPacket *componentParent=0, bool setLabels=true)
 Splits a disconnected triangulation into many smaller triangulations, one for each component. More...
 
unsigned long connectedSumDecomposition (NPacket *primeParent=0, bool setLabels=true)
 Splits this triangulation into its connected sum decomposition. More...
 
bool isThreeSphere () const
 Determines whether this is a triangulation of a 3-sphere. More...
 
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. More...
 
bool isBall () const
 Determines whether this is a triangulation of a 3-dimensional ball. More...
 
bool knowsBall () const
 Is it already known (or trivial to determine) whether or not this is a triangulation of a 3-dimensional ball? See isBall() for further details. More...
 
NPacketmakeZeroEfficient ()
 Converts this into a 0-efficient triangulation of the same underlying 3-manifold. More...
 
bool isSolidTorus () const
 Determines whether this is a triangulation of the solid torus; that is, the unknot complement. More...
 
bool knowsSolidTorus () const
 Is it already known (or trivial to determine) whether or not this is a triangulation of a solid torus (that is, the unknot complement)? See isSolidTorus() for further details. More...
 
bool isIrreducible () const
 Determines whether the underlying 3-manifold (which must be closed) is irreducible. More...
 
bool knowsIrreducible () const
 Is it already known (or trivial to determine) whether or not the underlying 3-manifold is irreducible? See isIrreducible() for further details. More...
 
bool hasCompressingDisc () const
 Searches for a compressing disc within the underlying 3-manifold. More...
 
bool knowsCompressingDisc () const
 Is it already known (or trivial to determine) whether or not the underlying 3-manifold contains a compressing disc? See hasCompressingDisc() for further details. More...
 
bool isHaken () const
 Determines whether the underlying 3-manifold (which must be closed and orientable) is Haken. More...
 
bool knowsHaken () const
 Is it already known (or trivial to determine) whether or not the underlying 3-manifold is Haken? See isHaken() for further details. More...
 
bool hasSimpleCompressingDisc () const
 Searches for a "simple" compressing disc inside this triangulation. More...
 
Subdivisions, Extensions and Covers
void makeDoubleCover ()
 Converts this triangulation into its double cover. More...
 
bool idealToFinite (bool forceDivision=false)
 Converts an ideal triangulation into a finite triangulation. More...
 
bool finiteToIdeal ()
 Converts each real boundary component into a cusp (i.e., an ideal vertex). More...
 
void barycentricSubdivision ()
 Does a barycentric subdivision of the triangulation. More...
 
void drillEdge (NEdge *e)
 Drills out a regular neighbourhood of the given edge of the triangulation. More...
 
Building Triangulations
NTetrahedronlayerOn (NEdge *edge)
 Performs a layering upon the given boundary edge of the triangulation. More...
 
NTetrahedroninsertLayeredSolidTorus (unsigned long cuts0, unsigned long cuts1)
 Inserts a new layered solid torus into the triangulation. More...
 
void insertLayeredLensSpace (unsigned long p, unsigned long q)
 Inserts a new layered lens space L(p,q) into the triangulation. More...
 
void insertLayeredLoop (unsigned long length, bool twisted)
 Inserts a layered loop of the given length into this triangulation. More...
 
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. More...
 
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. More...
 
void insertTriangulation (const NTriangulation &source)
 Inserts a copy of the given triangulation into this triangulation. More...
 
bool insertRehydration (const std::string &dehydration)
 Inserts the rehydration of the given string into this triangulation. More...
 
void insertConstruction (unsigned long nTetrahedra, const int adjacencies[][4], const int gluings[][4][4])
 Inserts into this triangulation a set of tetrahedra and their gluings as described by the given integer arrays. More...
 
Exporting Triangulations
std::string dehydrate () const
 Dehydrates this triangulation into an alphabetical string. More...
 
std::string isoSig (NIsomorphism **relabelling=0) const
 Constructs the isomorphism signature for this triangulation. More...
 
std::string dumpConstruction () const
 Returns C++ code that can be used with insertConstruction() to reconstruct this triangulation. More...
 
std::string snapPea () const
 Returns a string containing the full contents of a SnapPea data file that describes this triangulation. More...
 
- Public Member Functions inherited from regina::NPacket
 NPacket (NPacket *parent=0)
 Constructor that inserts the new packet into the overall tree structure. More...
 
virtual ~NPacket ()
 Destructor that also orphans this packet and destroys all of its descendants. More...
 
virtual PacketType getPacketType () const =0
 Returns the unique integer ID representing this type of packet. More...
 
virtual std::string getPacketTypeName () const =0
 Returns an English name for this type of packet. More...
 
const std::string & getPacketLabel () const
 Returns the label associated with this individual packet. More...
 
std::string getHumanLabel () const
 Returns the label associated with this individual packet, adjusted if necessary for human-readable output. More...
 
void setPacketLabel (const std::string &newLabel)
 Sets the label associated with this individual packet. More...
 
std::string getFullName () const
 Returns a descriptive text string for the packet. More...
 
std::string makeUniqueLabel (const std::string &base) const
 Returns a new label that cannot be found anywhere in the entire tree structure. More...
 
bool makeUniqueLabels (NPacket *reference)
 Ensures that all packet labels in both this and the given packet tree combined are distinct. More...
 
bool hasTag (const std::string &tag) const
 Determines whether this packet has the given associated tag. More...
 
bool hasTags () const
 Determines whether this packet has any associated tags at all. More...
 
bool addTag (const std::string &tag)
 Associates the given tag with this packet. More...
 
bool removeTag (const std::string &tag)
 Removes the association of the given tag with this packet. More...
 
void removeAllTags ()
 Removes all associated tags from this packet. More...
 
const std::set< std::string > & getTags () const
 Returns the set of all tags associated with this packet. More...
 
bool listen (NPacketListener *listener)
 Registers the given packet listener to listen for events on this packet. More...
 
bool isListening (NPacketListener *listener)
 Determines whether the given packet listener is currently listening for events on this packet. More...
 
bool unlisten (NPacketListener *listener)
 Unregisters the given packet listener so that it no longer listens for events on this packet. More...
 
NPacketgetTreeParent () const
 Determines the parent packet in the tree structure. More...
 
NPacketgetFirstTreeChild () const
 Determines the first child of this packet in the tree structure. More...
 
NPacketgetLastTreeChild () const
 Determines the last child of this packet in the tree structure. More...
 
NPacketgetNextTreeSibling () const
 Determines the next sibling of this packet in the tree structure. More...
 
NPacketgetPrevTreeSibling () const
 Determines the previous sibling of this packet in the tree structure. More...
 
NPacketgetTreeMatriarch () const
 Determines the matriarch (the root) of the tree to which this packet belongs. More...
 
unsigned levelsDownTo (const NPacket *descendant) const
 Counts the number of levels between this packet and its given descendant in the tree structure. More...
 
unsigned levelsUpTo (const NPacket *ancestor) const
 Counts the number of levels between this packet and its given ancestor in the tree structure. More...
 
bool isGrandparentOf (const NPacket *descendant) const
 Determines if this packet is equal to or an ancestor of the given packet in the tree structure. More...
 
unsigned long getNumberOfChildren () const
 Returns the number of immediate children of this packet. More...
 
unsigned long getNumberOfDescendants () const
 Returns the total number of descendants of this packet. More...
 
unsigned long getTotalTreeSize () const
 Determines the total number of packets in the tree or subtree for which this packet is matriarch. More...
 
void insertChildFirst (NPacket *child)
 Inserts the given packet as the first child of this packet. More...
 
void insertChildLast (NPacket *child)
 Inserts the given packet as the last child of this packet. More...
 
void insertChildAfter (NPacket *newChild, NPacket *prevChild)
 Inserts the given packet as a child of this packet at the given location in this packet's child list. More...
 
void makeOrphan ()
 Cuts this packet away from its parent in the tree structure and instead makes it matriarch of its own tree. More...
 
void reparent (NPacket *newParent, bool first=false)
 Cuts this packet away from its parent in the tree structure, and inserts it as a child of the given packet instead. More...
 
void swapWithNextSibling ()
 Swaps this packet with its next sibling in the sequence of children beneath their common parent packet. More...
 
void moveUp (unsigned steps=1)
 Moves this packet the given number of steps towards the beginning of its sibling list. More...
 
void moveDown (unsigned steps=1)
 Moves this packet the given number of steps towards the end of its sibling list. More...
 
void moveToFirst ()
 Moves this packet to be the first in its sibling list. More...
 
void moveToLast ()
 Moves this packet to be the last in its sibling list. More...
 
void sortChildren ()
 Sorts the immediate children of this packet according to their packet labels. More...
 
NPacketnextTreePacket ()
 Finds the next packet after this in a complete depth-first iteration of the entire tree structure to which this packet belongs. More...
 
const NPacketnextTreePacket () const
 Finds the next packet after this in a complete depth-first iteration of the entire tree structure to which this packet belongs. More...
 
NPacketfirstTreePacket (const std::string &type)
 Finds the first packet of the requested type in a complete depth-first iteration of the tree structure. More...
 
const NPacketfirstTreePacket (const std::string &type) const
 Finds the first packet of the requested type in a complete depth-first iteration of the tree structure. More...
 
NPacketnextTreePacket (const std::string &type)
 Finds the next packet after this of the requested type in a complete depth-first iteration of the entire tree structure. More...
 
const NPacketnextTreePacket (const std::string &type) const
 Finds the next packet after this of the requested type in a complete depth-first iteration of the entire tree structure. More...
 
NPacketfindPacketLabel (const std::string &label)
 Finds the packet with the requested label in the tree or subtree for which this packet is matriarch. More...
 
const NPacketfindPacketLabel (const std::string &label) const
 Finds the packet with the requested label in the tree or subtree for which this packet is matriarch. More...
 
bool isPacketEditable () const
 Determines whether this packet can be altered without invalidating or otherwise upsetting any of its immediate children. More...
 
NPacketclone (bool cloneDescendants=false, bool end=true) const
 Clones this packet (and possibly its descendants), assigns to it a suitable unused label and inserts the clone into the tree as a sibling of this packet. More...
 
void writeXMLFile (std::ostream &out) const
 Writes a complete XML file containing the subtree with this packet as matriarch. More...
 
std::string internalID () const
 Returns a unique string ID that identifies this packet. More...
 
- Public Member Functions inherited from regina::ShareableObject
 ShareableObject ()
 Default constructor that does nothing. More...
 
virtual ~ShareableObject ()
 Default destructor that does nothing. More...
 
std::string str () const
 Returns the output from writeTextShort() as a string. More...
 
std::string toString () const
 A deprecated alias for str(), which returns the output from writeTextShort() as a string. More...
 
std::string detail () const
 Returns the output from writeTextLong() as a string. More...
 
std::string toStringLong () const
 A deprecated alias for detail(), which returns the output from writeTextLong() as a string. More...
 

Static Public Member Functions

static NXMLPacketReadergetXMLReader (NPacket *parent, NXMLTreeResolver &resolver)
 
Importing Triangulations
static NTriangulationenterTextTriangulation (std::istream &in, std::ostream &out)
 Allows the user to interactively enter a triangulation in plain text. More...
 
static NTriangulationrehydrate (const std::string &dehydration)
 Rehydrates the given alphabetical string into a new triangulation. More...
 
static NTriangulationfromIsoSig (const std::string &signature)
 Recovers a full triangulation from an isomorphism signature. More...
 
static NTriangulationfromSnapPea (const std::string &snapPeaData)
 Extracts a triangulation from a string that contains the full contents of a SnapPea data file. More...
 
- Static Public Member Functions inherited from regina::NPacket
static NXMLPacketReadergetXMLReader (NPacket *parent, NXMLTreeResolver &resolver)
 Returns a newly created XML element reader that will read the contents of a single XML packet element. More...
 

Protected Member Functions

virtual NPacketinternalClonePacket (NPacket *parent) const
 Makes a newly allocated copy of this packet. More...
 
virtual void writeXMLPacketData (std::ostream &out) const
 Writes a chunk of XML containing the data for this packet only. More...
 
void cloneFrom (const NTriangulation &from)
 Turns this triangulation into a clone of the given triangulation. More...
 
- Protected Member Functions inherited from regina::NPacket
void writeXMLPacketTree (std::ostream &out) const
 Writes a chunk of XML containing the subtree with this packet as matriarch. More...
 

Friends

class regina::NTetrahedron
 
class regina::NXMLTriangulationReader
 

Additional Inherited Members

- Static Protected Member Functions inherited from regina::NGenericTriangulation< 3 >
static std::string isoSig (const typename DimTraits< dim >::Triangulation &tri, typename DimTraits< dim >::Isomorphism **relabelling=0)
 Constructs the isomorphism signature for the given triangulation. More...
 
static DimTraits< dim >
::Triangulation
fromIsoSig (const std::string &sig)
 Recovers a full triangulation from an isomorphism signature. More...
 
static size_t isoSigComponentSize (const std::string &sig)
 Deduces the number of top-dimensional simplices in a connected triangulation from its isomorphism signature. More...
 

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.

Triangles, 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.

The management of tetrahedra within a triangulation has become simpler and safer as of Regina 4.90. In older versions (Regina 4.6 and earlier), users were required to create tetrahedra, individually add them to triangulations, and manually notify a triangulation whenever its tetrahedron gluings changed. As of Regina 4.90, new tetrahedra are created using NTriangulation::newTetrahedron() which automatically places them within a triangulation, and all gluing changes are likewise communicated to the triangulation automatically. These are part of a larger suite of changes (all designed to help the user avoid inconsistent states and accidental crashes); see the NTetrahedron class notes for further details.

Test:
Included in the test suite.
Todo:

Feature: Is the boundary incompressible?

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.

Member Typedef Documentation

Used to iterate through boundary components.

typedef std::vector<NComponent*>::const_iterator regina::NTriangulation::ComponentIterator

Used to iterate through components.

typedef std::vector<NEdge*>::const_iterator regina::NTriangulation::EdgeIterator

Used to iterate through edges.

typedef std::vector<NTriangle*>::const_iterator regina::NTriangulation::FaceIterator

A deprecated alias for TriangleIterator.

typedef std::vector<NTetrahedron*>::const_iterator regina::NTriangulation::TetrahedronIterator

Used to iterate through tetrahedra.

typedef std::vector<NTriangle*>::const_iterator regina::NTriangulation::TriangleIterator

Used to iterate through triangles.

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

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

typedef std::vector<NVertex*>::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
cloneMethe triangulation to clone.
regina::NTriangulation::NTriangulation ( const std::string &  description)

"Magic" constructor that tries to find some way to interpret the given string as a triangulation.

At present, Regina understands the following types of strings (and attempts to parse them in the following order):

This list may grow in future versions of Regina.

Regina will also set the packet label accordingly.

If Regina cannot interpret the given string, this will be left as the empty triangulation.

Parameters
descriptiona string that describes a 3-manifold triangulation.
regina::NTriangulation::~NTriangulation ( )
inlinevirtual

Destroys this triangulation.

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

Member Function Documentation

void regina::NTriangulation::addTetrahedron ( NTetrahedron tet)

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.

Precondition
The given tetrahedron does not already belong to a different triangulation (though already belonging to this triangulation is perfectly fine).
Deprecated:
Users should create tetrahedra by calling newTetrahedron() or newTetrahedron(const std::string&), which will add the tetrahedron to the triangulation automatically.
Warning
As of Regina 4.90, this routine will also add any neighbouring tetrahedra that do not yet belong to a triangulation; moreover, this addition is recursive. This is done to ensure that, whenever one tetrahedron belongs to a triangulation, everything that it is joined to (directly or indirectly) also belongs to that same triangulation. See the NTetrahedron class notes for further details on how tetrahedron management has changed in Regina 4.90 and above.
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
tetthe 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 triangle and the midpoint of each edge.

Author
David Letscher
long regina::NTriangulation::boundaryComponentIndex ( const NBoundaryComponent bc) const
inline

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

This routine was introduced in Regina 4.5, and replaces the old getBoundaryComponentIndex(). The name has been changed because, unlike the old routine, it requires that the given boundary component belongs to the triangulation (a consequence of some significant memory optimisations).

Precondition
The given boundary component belongs to this triangulation.
Warning
Passing a null pointer to this routine will probably crash your program.
Parameters
bcspecifies 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.
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
fromthe triangulation from which this triangulation will be cloned.
bool regina::NTriangulation::closeBook ( NEdge e,
bool  check = true,
bool  perform = true 
)

Checks the eligibility of and/or performs a book closing move about the given boundary edge.

This involves taking a boundary edge of the triangulation and folding together the two boundary triangles on either side. This move is the inverse of the openBook() move, and is used to simplify the boundary of the triangulation. This move can be done if:

  • the edge e is a boundary edge;
  • the two boundary triangles that it joins are distinct;
  • the two vertices opposite e in each of these boundary triangles are valid and distinct;
  • if edges e1 and e2 of one boundary triangle are to be folded onto edges f1 and f2 of the other boundary triangle respectively, then we do not have both e1 = e2 and f1 = f2.

There are in fact several other "distinctness" conditions on the edges e1, e2, f1 and f2, but they follow automatically from the "distinct vertices" condition above.

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 (triangles, components, etc.) will be reconstructed, which means any pointers to old skeletal objects (such as the argument f) can no longer be used.

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 given edge is an edge of this triangulation.
Parameters
ethe edge about which to perform the move.
checktrue if we are to check whether the move is allowed (defaults to true).
performtrue 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::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 (triangles, components, etc.) will be reconstructed, which means any pointers to old skeletal objects (such as the argument e) can no longer be used.

The eligibility requirements for this move are somewhat involved, and are discussed in detail in the collapseEdge() source code for those who are interested.

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 given edge is an edge of this triangulation.
Parameters
ethe edge to collapse.
checktrue if we are to check whether the move is allowed (defaults to true).
performtrue 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.
long regina::NTriangulation::componentIndex ( const NComponent component) const
inline

Returns the index of the given component in the triangulation.

This routine was introduced in Regina 4.5, and replaces the old getComponentIndex(). The name has been changed because, unlike the old routine, it requires that the given component belongs to the triangulation (a consequence of some significant memory optimisations).

Precondition
The given component belongs to this triangulation.
Warning
Passing a null pointer to this routine will probably crash your program.
Parameters
componentspecifies 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.
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
primeParentthe packet beneath which the new prime summand triangulations will be inserted, or 0 if they should be inserted directly beneath this triangulation.
setLabelstrue 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.
std::string regina::NTriangulation::dehydrate ( ) const

Dehydrates this triangulation into an alphabetical string.

A dehydration string is a compact text representation of a triangulation, introduced by Callahan, Hildebrand and Weeks for their cusped hyperbolic census (see below). The dehydration string of an n-tetrahedron triangulation consists of approximately (but not precisely) 5n/2 lower-case letters.

Dehydration strings come with some restrictions:

  • They rely on the triangulation being "canonical" in some combinatorial sense. This is not enforced here; instead a combinatorial isomorphism is applied to make the triangulation canonical, and this isomorphic triangulation is dehydrated instead. Note that the original triangulation is not changed.
  • They require the triangulation to be connected.
  • They require the triangulation to have no boundary triangles (though ideal triangulations are fine).
  • They can only support triangulations with at most 25 tetrahedra.

The routine rehydrate() can be used to recover a triangulation from a dehydration string. Note that the triangulation recovered might not be identical to the original, but it is guaranteed to be an isomorphic copy.

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.

Returns
a dehydrated representation of this triangulation (or an isomorphic variant of this triangulation), or the empty string if dehydration is not possible because the triangulation is disconnected, has boundary triangles or contains too many tetrahedra.
See Also
rehydrate
insertRehydration
bool regina::NTriangulation::dependsOnParent ( ) const
inlinevirtual

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.

void regina::NTriangulation::drillEdge ( NEdge e)

Drills out a regular neighbourhood of the given edge of the triangulation.

This is done by (i) performing two barycentric subdivisions, (ii) removing all tetrahedra that touch the original edge, and (iii) simplifying the resulting triangulation.

Warning
The second barycentric subdivision will multiply the number of tetrahedra by 576; as a result this routine might be slow, and the number of tetrahedra at the end might be large (even taking the simplification into account).
Parameters
ethe edge to drill out.
std::string regina::NTriangulation::dumpConstruction ( ) const

Returns C++ code that can be used with insertConstruction() to reconstruct this triangulation.

The code produced will consist of the following:

  • the declaration and initialisation of two integer arrays, describing the tetrahedron gluings in this trianguation;
  • two additional lines that declare a new NTriangulation and call insertConstruction() to rebuild this triangulation.

The main purpose of this routine is to generate the two integer arrays, which can be tedious and error-prone to code up by hand.

Note that the number of lines of code produced grows linearly with the number of tetrahedra. If this triangulation is very large, the returned string will be very large as well.

Returns
the C++ code that was generated.
long regina::NTriangulation::edgeIndex ( const NEdge edge) const
inline

Returns the index of the given edge in the triangulation.

This routine was introduced in Regina 4.5, and replaces the old getEdgeIndex(). The name has been changed because, unlike the old routine, it requires that the given edge belongs to the triangulation (a consequence of some significant memory optimisations).

Precondition
The given edge belongs to this triangulation.
Warning
Passing a null pointer to this routine will probably crash your program.
Parameters
edgespecifies 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.
static 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
inthe input stream from which text will be read.
outthe output stream to which prompts will be written.
Returns
the triangulation entered in by the user.
long regina::NTriangulation::faceIndex ( const NTriangle triangle) const
inline

A deprecated alias for triangleIndex().

This routine returns the index of the given triangle in the triangulation. See triangleIndex() for further details.

Deprecated:
This routine will be removed in a future version of Regina. Please use triangleIndex() instead.
Parameters
trianglespecifies which triangle to find in the triangulation.
Returns
the index of the specified triangle, where 0 is the first triangle, 1 is the second and so on.
unsigned long regina::NTriangulation::findAllSubcomplexesIn ( const NTriangulation other,
std::list< NIsomorphism * > &  results 
) const

Finds all ways in which an isomorphic copy of this triangulation is contained within the given triangulation, possibly as a subcomplex of some larger component (or components).

This routine behaves identically to isContainedIn(), except that instead of returning just one isomorphism (which may be boundary incomplete and need not be onto), all such isomorphisms are returned.

See the isContainedIn() notes for additional information.

The isomorphisms that are found will be inserted into the given list. These isomorphisms will be newly created, and the caller of this routine is responsible for destroying them. The given list will not be emptied before the new isomorphisms are inserted.

Python:
Not present.
Parameters
otherthe triangulation in which to search for isomorphic copies of this triangulation.
resultsthe list in which any isomorphisms found will be stored.
Returns
the number of isomorphisms that were found.
bool regina::NTriangulation::finiteToIdeal ( )

Converts each real boundary component into a cusp (i.e., an ideal vertex).

Only boundary components formed from real tetrahedron faces will be affected; ideal boundary components are already cusps and so will not be changed.

One side-effect of this operation is that all spherical boundary components will be filled in with balls.

This operation is performed by attaching a new tetrahedron to each boundary triangle and then gluing these new tetrahedra together in a way that mirrors the adjacencies of the underlying boundary triangles. Each boundary component will thereby be pushed up through the new tetrahedra and converted into a cusp formed using vertices of these new tetrahedra.

Note that this operation is a loose converse of idealToFinite().

Warning
If a real boundary component contains vertices whose links are not discs, this operation may have unexpected results.
Returns
true if changes were made, or false if the original triangulation contained no real boundary components.
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 (i) the edge is valid and non-boundary, and (ii) 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 (triangles, components, etc.) will be reconstructed, which means any pointers to old skeletal objects (such as the argument e) can no longer be used.

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 given edge is an edge of this triangulation.
Parameters
ethe edge about which to perform the move.
newAxisSpecifies 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
  1. 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.
checktrue if we are to check whether the move is allowed (defaults to true).
performtrue 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.
NTriangulation * regina::NTriangulation::fromIsoSig ( const std::string &  signature)
inlinestatic

Recovers a full triangulation from an isomorphism signature.

See isoSig() for more information on isomorphism signatures.

The triangulation that is returned will be newly created.

Calling isoSig() followed by fromIsoSig() is not guaranteed to produce an identical triangulation to the original, but it is guaranteed to produce a combinatorially isomorphic triangulation.

For a full and precise description of the isomorphism signature format, see Simplification paths in the Pachner graphs of closed orientable 3-manifold triangulations, Burton, 2011, arXiv:1110.6080.

Parameters
signaturethe isomorphism signature of the triangulation to construct. Note that, unlike dehydration strings, case is important for isomorphism signatures.
Returns
a newly allocated triangulation if the reconstruction was successful, or null if the given string was not a valid isomorphism signature.
static NTriangulation* regina::NTriangulation::fromSnapPea ( const std::string &  snapPeaData)
static

Extracts a triangulation from a string that contains the full contents of a SnapPea data file.

This routine could, for instance, be used to receive a triangulation from SnapPy without writing to the filesystem.

If you wish to read a triangulation from a SnapPea file, you should use the global function readSnapPea() instead (which has better performance, and does not require you to construct an enormous intermediate string).

For details on how the triangulation will be extracted, see the documentation for readSnapPea().

The triangulation that is returned will be newly created. If the SnapPea data is not in the correct format, this routine will return 0 instead.

Precondition
The first two lines of the SnapPea file each contain at most 1000 characters.
Parameters
snapPeaDataa string containing the full contents of a SnapPea data file.
Returns
a new triangulation extracted from the given data, or 0 on error.
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
indexthe index of the desired boundary component, ranging from 0 to getNumberOfBoundaryComponents()-1 inclusive.
Returns
the requested boundary component.
const std::vector< NBoundaryComponent * > & 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
indexthe index of the desired component, ranging from 0 to getNumberOfComponents()-1 inclusive.
Returns
the requested component.
const std::vector< NComponent * > & 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 edge in this triangulation.

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
indexthe index of the desired edge, ranging from 0 to getNumberOfEdges()-1 inclusive.
Returns
the requested edge.
const std::vector< NEdge * > & 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

A deprecated alias for getEulerCharTri().

This routine calculates the Euler characteristic of this triangulation. Since it treats cusps in a non-standard way, it was renamed to getEulerCharTri() in Regina 4.4 to clarify that this might differ from the Euler characteristic of the corresponding compact manifold.

See getEulerCharTri() for further details.

Deprecated:
This routine will be removed in a future version of Regina. Please use getEulerCharTri() instead.
Returns
the Euler characteristic of this triangulation.
long regina::NTriangulation::getEulerCharManifold ( ) const

Returns the Euler characteristic of the corresponding compact 3-manifold.

Instead of simply calculating V-E+F-T, this routine also:

  • treats ideal vertices as surface boundary components (i.e., effectively truncates them);
  • truncates invalid boundary vertices (i.e., boundary vertices whose links are not discs);
  • truncates the projective plane cusps at the midpoints of invalid edges (edges identified with themselves in reverse).

For ideal triangulations, this routine therefore computes the proper Euler characteristic of the manifold (unlike getEulerCharTri(), which does not).

For triangulations whose vertex links are all spheres or discs, this routine and getEulerCharTri() give identical results.

Returns
the Euler characteristic of the corresponding compact manifold.
long regina::NTriangulation::getEulerCharTri ( ) const
inline

Returns the Euler characteristic of this triangulation.

This will be evaluated strictly as V-E+F-T.

Note that this routine handles cusps in a non-standard way. Since it computes the Euler characteristic of the triangulation (and not the underlying manifold), this routine will treat each cusp as a single vertex, and not as a surface boundary component.

For a routine that handles cusps properly (i.e., treats them as surface boundary components when computing the Euler characteristic), see getEulerCharManifold() instead.

This routine was previously called getEulerCharacteristic() in Regina 4.3.1 and earlier. It was renamed in Regina 4.4 to clarify the non-standard handling of cusps.

Returns
the Euler characteristic of this triangulation.
NTriangle * regina::NTriangulation::getFace ( unsigned long  index) const
inline

A deprecated alias for getTriangle().

This routine returns the requested triangular face in the triangulation. See getTriangle() for further details.

Deprecated:
This routine will be removed in a future version of Regina. Please use getTriangle() instead.
Parameters
indexthe index of the desired triangle, ranging from 0 to getNumberOfTriangles()-1 inclusive.
Returns
the requested triangle.
const std::vector< NTriangle * > & regina::NTriangulation::getFaces ( ) const
inline

A deprecated alias for getTriangles().

This routine returns all triangular faces in this triangulation. See getTriangles() for further details.

Deprecated:
This routine will be removed in a future version of Regina. Please use getTriangles() instead.
Returns
the list of all triangles.
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 reference that is returned from this routine 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 reference that is returned from this routine 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 reference that is returned from this routine 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 reference that is returned from this routine 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 reference that is returned from this routine 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< 3 > ( ) const
inline

A deprecated alias for getNumberOfTriangles().

This routine returns the number of triangular faces in this triangulation. See getNumberOfTriangles() for further details.

Do not confuse this deprecated alias with the (non-deprecated) tempate function getNumberOfFaces<dim>().

Deprecated:
This routine will be removed in a future version of Regina. Please use getNumberOfTriangles() instead.
Returns
the number of triangles.
template<int dim>
unsigned long regina::NTriangulation::getNumberOfFaces ( ) const

Returns the number of faces of the given dimension in this triangulation.

This template function is to assist with writing dimension-agnostic code that can be reused to work in different dimensions.

Precondition
the template argument dim is between 0 and 3 inclusive.
Python:
Not present.
Returns
the number of faces of the given dimension.
unsigned long regina::NTriangulation::getNumberOfSimplices ( ) const
inline

A dimension-agnostic alias for getNumberOfTetrahedra().

This is to assist with writing dimension-agnostic code that can be reused to work in different dimensions.

Here "simplex" refers to a top-dimensional simplex (which for 3-manifold triangulations means a tetrahedron).

See getNumberOfTetrahedra() for further information.

unsigned long regina::NTriangulation::getNumberOfTetrahedra ( ) const
inline

Returns the number of tetrahedra in the triangulation.

Returns
the number of tetrahedra.
unsigned long regina::NTriangulation::getNumberOfTriangles ( ) const
inline

Returns the number of triangular faces in this triangulation.

Returns
the number of triangles.
unsigned long regina::NTriangulation::getNumberOfVertices ( ) const
inline

Returns the number of vertices in this triangulation.

Returns
the number of vertices.
NTetrahedron * regina::NTriangulation::getSimplex ( unsigned long  index)
inline

A dimension-agnostic alias for getTetrahedron().

This is to assist with writing dimension-agnostic code that can be reused to work in different dimensions.

Here "simplex" refers to a top-dimensional simplex (which for 3-manifold triangulations means a tetrahedron).

See getTetrahedron() for further information.

const NTetrahedron * regina::NTriangulation::getSimplex ( unsigned long  index) const
inline

A dimension-agnostic alias for getTetrahedron().

This is to assist with writing dimension-agnostic code that can be reused to work in different dimensions.

Here "simplex" refers to a top-dimensional simplex (which for 3-manifold triangulations means a tetrahedron).

See getTetrahedron() for further information.

const std::vector< NTetrahedron * > & regina::NTriangulation::getSimplices ( ) const
inline

A dimension-agnostic alias for getTetrahedra().

This is to assist with writing dimension-agnostic code that can be reused to work in different dimensions.

Here "simplex" refers to a top-dimensional simplex (which for 3-manifold triangulations means a tetrahedron).

See getTetrahedra() for further information.

const std::vector< NTetrahedron * > & 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.
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.

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

Parameters
indexspecifies which tetrahedron to return; this value should be between 0 and getNumberOfTetrahedra()-1 inclusive.
Returns
the indexth tetrahedron in the triangulation.
NTriangle * regina::NTriangulation::getTriangle ( unsigned long  index) const
inline

Returns the requested triangular face in this triangulation.

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

Parameters
indexthe index of the desired triangle, ranging from 0 to getNumberOfTriangles()-1 inclusive.
Returns
the requested triangle.
const std::vector< NTriangle * > & regina::NTriangulation::getTriangles ( ) const
inline

Returns all triangular faces of this triangulation.

Bear in mind that each time the triangulation changes, the triangles 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 triangles.
NVertex * regina::NTriangulation::getVertex ( unsigned long  index) const
inline

Returns the requested vertex in this triangulation.

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
indexthe index of the desired vertex, ranging from 0 to getNumberOfVertices()-1 inclusive.
Returns
the requested vertex.
const std::vector< NVertex * > & 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.
void regina::NTriangulation::gluingsHaveChanged ( )
inline

This routine now does nothing, and should not be used.

Deprecated:
In Regina versions 4.6 and earlier, this routine was used to manually notify the triangulation that the gluings of tetrahedra had changed. In Regina 4.90 and later this notification is automatic. This routine now does nothing at all, and can safely be removed from any existing code.
bool regina::NTriangulation::hasBoundaryFaces ( ) const
inline

A deprecated alias for hasBoundaryTriangles().

This routine determines whether this triangulation has any boundary triangles. See hasBoundaryTriangles() for further details.

Deprecated:
This routine will be removed in a future version of Regina. Please use hasBoundaryTriangles() instead.
Returns
true if and only if there are boundary triangles.
bool regina::NTriangulation::hasBoundaryTriangles ( ) const
inline

Determines if this triangulation has any boundary triangles.

Returns
true if and only if there are boundary triangles.
bool regina::NTriangulation::hasCompressingDisc ( ) const

Searches for a compressing disc within the underlying 3-manifold.

Let M be the underlying 3-manifold and let B be its boundary. By a compressing disc, we mean a disc D properly embedded in M, where the boundary of D lies in B but does not bound a disc in B.

This routine will first call the heuristic routine hasSimpleCompressingDisc() in the hope of obtaining a fast answer. If this fails, it will do one of two things:

  • If the triangulation is orientable and 1-vertex, it will use the linear programming and crushing machinery outlined in "Computing closed essential surfaces in knot complements", Burton, Coward and Tillmann, SCG '13, p405-414, 2013. This is often extremely fast, even for triangulations with many tetrahedra.
  • If the triangulation is non-orientable or has multiple vertices then it will run a full enumeration of vertex normal surfaces, as described in "Algorithms for the complete decomposition of a closed 3-manifold", Jaco and Tollefson, Illinois J. Math. 39 (1995), 358-406. As the number of tetrahedra grows, this can become extremely slow.

This routine will work on a copy of this triangulation, not the original. In particular, the copy will be simplified, which means that there is no harm in calling this routine on an unsimplified triangulation.

If this triangulation has no boundary components, this routine will simply return false.

Precondition
This triangulation is valid and is not ideal.
The underlying 3-manifold is irreducible.
Warning
This routine can be infeasibly slow for large triangulations (particularly those that are non-orientable or have multiple vertices), since it may need to perform a full enumeration of vertex normal surfaces, and since it might perform "large" operations on these surfaces such as cutting along them. See hasSimpleCompressingDisc() for a "heuristic shortcut" that is faster but might not give a definitive answer.
Returns
true if the underlying 3-manifold contains a compressing disc, or false if it does not.
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.
NNormalSurface* regina::NTriangulation::hasNonTrivialSphereOrDisc ( )

Searches for a non-vertex-linking normal sphere or disc within this triangulation.

If such a surface exists within this triangulation, this routine is guaranteed to find one.

Note that the surface returned (if any) depends upon this triangulation, and so the surface must be destroyed before this triangulation is destroyed.

Warning
This routine may, in some scenarios, temporarily modify the packet tree by creating and then destroying a normal surface list.
Returns
a newly allocated non-vertex-linking normal sphere or disc, or 0 if none exists.
NNormalSurface* regina::NTriangulation::hasOctagonalAlmostNormalSphere ( )

Searches for an octagonal almost normal 2-sphere within this triangulation.

If such a surface exists, this routine is guaranteed to find one.

Note that the surface returned (if any) depends upon this triangulation, and so the surface must be destroyed before this triangulation is destroyed.

Precondition
This triangulation is valid, closed, orientable, connected, and 0-efficient. These preconditions are almost certainly more restrictive than they need to be, but we stay safe for now.
Warning
This routine may, in some scenarios, temporarily modify the packet tree by creating and then destroying a normal surface list.
Returns
a newly allocated non-vertex-linking normal sphere or disc, or 0 if none exists.
bool regina::NTriangulation::hasSimpleCompressingDisc ( ) const

Searches for a "simple" compressing disc inside this triangulation.

Let M be the underlying 3-manifold and let B be its boundary. By a compressing disc, we mean a disc D properly embedded in M, where the boundary of D lies in B but does not bound a disc in B.

By a simple compressing disc, we mean a compressing disc that has a very simple combinatorial structure (here "simple" is subject to change; see the warning below). Examples include the compressing disc inside a 1-tetrahedron solid torus, or a compressing disc formed from a single internal triangle surrounded by three boundary edges.

The purpose of this routine is to avoid working with normal surfaces within a large triangulation where possible. This routine is relatively fast, and if it returns true then this 3-manifold definitely contains a compressing disc. If this routine returns false then there might or might not be a compressing disc; the user will need to perform a full normal surface enumeration using hasCompressingDisc() to be sure.

This routine will work on a copy of this triangulation, not the original. In particular, the copy will be simplified, which means that there is no harm in calling this routine on an unsimplified triangulation.

If this triangulation has no boundary components, this routine will simply return false.

For further information on this test, see "The Weber-Seifert dodecahedral space is non-Haken", Benjamin A. Burton, J. Hyam Rubinstein and Stephan Tillmann, Trans. Amer. Math. Soc. 364:2 (2012), pp. 911-932.

Warning
The definition of "simple" is subject to change in future releases of Regina. That is, this routine may be expanded over time to identify more types of compressing discs (thus making it more useful as a "heuristic shortcut").
Precondition
This triangulation is valid and is not ideal.
Returns
true if a simple compressing disc was found, or false if not. Note that even with a return value of false, there might still be a compressing disc (just not one with a simple combinatorial structure).
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.

Note that this operation is a loose converse of finiteToIdeal().

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
forceDivisionspecifies 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
a1a parameter describing the first layered solid torus in the augmented triangular solid torus; this may be either positive or negative.
b1a parameter describing the first layered solid torus in the augmented triangular solid torus; this may be either positive or negative.
a2a parameter describing the second layered solid torus in the augmented triangular solid torus; this may be either positive or negative.
b2a parameter describing the second layered solid torus in the augmented triangular solid torus; this may be either positive or negative.
a3a parameter describing the third layered solid torus in the augmented triangular solid torus; this may be either positive or negative.
b3a parameter describing the third layered solid torus in the augmented triangular solid torus; this may be either positive or negative.
void regina::NTriangulation::insertConstruction ( unsigned long  nTetrahedra,
const int  adjacencies[][4],
const int  gluings[][4][4] 
)

Inserts into this triangulation a set of tetrahedra and their gluings as described by the given integer arrays.

This routine is provided to make it easy to hard-code a medium-sized triangulation in a C++ source file. All of the pertinent data can be hard-coded into a pair of integer arrays at the beginning of the source file, avoiding an otherwise tedious sequence of many joinTo() calls.

An additional nTetrahedra tetrahedra will be inserted into this triangulation. The relationships between these tetrahedra should be stored in the two arrays as follows. Note that the new tetrahedra are numbered from 0 to (nTetrahedra - 1), and individual tetrahedron faces are numbered from 0 to 3.

The adjacencies array describes which tetrahedron faces are joined to which others. Specifically, adjacencies[t][f] should contain the number of the tetrahedron joined to face f of tetrahedron t. If this face is to be left as a boundary triangle, adjacencies[t][f] should be -1.

The gluings array describes the particular gluing permutations used when joining these tetrahedron faces together. Specifically, gluings[t][f][0..3] should describe the permutation used to join face f of tetrahedron t to its adjacent tetrahedron. These four integers should be 0, 1, 2 and 3 in some order, so that gluings[t][f][i] contains the image of i under this permutation. If face f of tetrahedron t is to be left as a boundary triangle, gluings[t][f][0..3] may contain anything (and will be duly ignored).

It is the responsibility of the caller of this routine to ensure that the given arrays are correct and consistent. No error checking will be performed by this routine.

Note that, for an existing triangulation, dumpConstruction() will output a pair of C++ arrays that can be copied into a source file and used to reconstruct the triangulation via this routine.

Python:
Not present.
Parameters
nTetrahedrathe number of additional tetrahedra to insert.
adjacenciesdescribes which of the new tetrahedron faces are to be identified. This array must have initial dimension at least nTetrahedra.
gluingsdescribes the specific gluing permutations by which these new tetrahedron faces should be identified. This array must also have initial dimension at least nTetrahedra.
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
pa parameter of the desired lens space.
qa 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
lengththe length of the new layered loop; this must be strictly positive.
twistedtrue 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
cuts0the smallest of the three desired intersection numbers.
cuts1the 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.

If you simply wish to convert a dehydration string into a new triangulation, use the static routine rehydrate() instead. See dehydrate() for more information on dehydration strings.

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

The routine dehydrate() can be used to extract a dehydration string from an existing triangulation. Dehydration followed by rehydration might not produce a triangulation identical to the original, but it is guaranteed to produce an isomorphic copy. See dehydrate() for the reasons behind this.

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
dehydrationa 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.
See Also
dehydrate
rehydrate
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
a1a parameter describing the first exceptional fibre.
b1a parameter describing the first exceptional fibre.
a2a parameter describing the second exceptional fibre.
b2a parameter describing the second exceptional fibre.
a3a parameter describing the third exceptional fibre.
b3a 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
sourcethe triangulation whose copy will be inserted.
bool regina::NTriangulation::intelligentSimplify ( )

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

This routine will attempt to reduce both the number of tetrahedra and the number of boundary triangles (with the number of tetrahedra as its priority).

Currently this routine uses simplifyToLocalMinimum() in combination with random 4-4 moves, book opening moves and book closing moves.

Warning
The specific behaviour of this routine may well change between releases.
Todo:
Optimise: Include random 2-3 moves to get out of wells.
Returns
true if and only if the triangulation was changed.
NPacket * regina::NTriangulation::internalClonePacket ( NPacket parent) const
inlineprotectedvirtual

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

Implements regina::NPacket.

bool regina::NTriangulation::isBall ( ) const

Determines whether this is a triangulation of a 3-dimensional ball.

This routine is based on isThreeSphere(), which in turn combines Rubinstein's 3-sphere recognition algorithm with 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 knowsBall() 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 triangulation of a 3-dimensional ball.
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 triangles of this triangulation need not correspond to boundary triangles 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.

If more than one such isomorphism exists, only one will be returned. For a routine that returns all such isomorphisms, see findAllSubcomplexesIn().

Parameters
otherthe 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::isHaken ( ) const

Determines whether the underlying 3-manifold (which must be closed and orientable) is Haken.

In other words, this routine determines whether the underlying 3-manifold contains an embedded closed two-sided incompressible surface.

Currently Hakenness testing is available only for irreducible manifolds. This routine will first test whether the manifold is irreducible and, if it is not, will return false immediately.

Precondition
This triangulation is valid, closed, orientable and connected.
Warning
This routine could be very slow for larger triangulations.
Returns
true if and only if the underlying 3-manifold is irreducible and Haken.
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.
bool regina::NTriangulation::isIrreducible ( ) const

Determines whether the underlying 3-manifold (which must be closed) is irreducible.

In other words, this routine determines whether every embedded sphere in the underlying 3-manifold bounds a ball.

If the underlying 3-manifold is orientable, this routine will use fast crushing and branch-and-bound methods. If the underlying 3-manifold is non-orientable, it will use a (much slower) full enumeration of vertex normal surfaces.

Warning
The algorithms used in this routine rely on normal surface theory and might be slow for larger triangulations.
Precondition
This triangulation is valid, closed, orientable and connected.
Returns
true if and only if the underlying 3-manifold is irreducible.
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
otherthe 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::isOrdered ( ) const

Determines if this triangulation is ordered; that is, if tetrahedron vertices are labelled so that all gluing permutations are order-preserving on the tetrahedron faces.

Equivalently, this tests whether the edges of the triangulation can all be oriented such that they induce a consistent ordering on the vertices of each tetrahedron.

Triangulations are not ordered by default, and indeed some cannot be ordered at all. The routine order() will attempt to relabel tetrahedron vertices to give an ordered triangulation.

Returns
true if and only if all gluing permutations are order preserving on the tetrahedron faces.
Author
Matthias Goerner
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::isOriented ( ) const

Determines if this triangulation is oriented; that is, if tetrahedron vertices are labelled in a way that preserves orientation across adjacent tetrahedron faces.

Specifically, this routine returns true if and only if every gluing permutation has negative sign.

Note that orientable triangulations are not always oriented by default. You can call orient() if you need the tetrahedra to be oriented consistently as described above.

A non-orientable triangulation can never be oriented.

Returns
true if and only if all tetrahedra are oriented consistently.
Author
Matthias Goerner
std::string regina::NTriangulation::isoSig ( NIsomorphism **  relabelling = 0) const
inline

Constructs the isomorphism signature for this triangulation.

An isomorphism signature is a compact text representation of a triangulation. Unlike dehydrations, an isomorphism signature uniquely determines a triangulation up to combinatorial isomorphism. That is, two triangulations are combinatorially isomorphic if and only if their isomorphism signatures are the same.

The isomorphism signature is constructed entirely of printable characters, and has length proportional to n log n, where n is the number of tetrahedra.

Isomorphism signatures are more general than dehydrations: they can be used with any triangulation (including closed, ideal, bounded, invalid and/or disconnected triangulations, as well as triangulations with large numbers of tetrahedra).

The time required to construct the isomorphism signature of a triangulation is O(n^2 log^2 n).

The routine fromIsoSig() can be used to recover a triangulation from an isomorphism signature. The triangulation recovered might not be identical to the original, but it will be combinatorially isomorphic.

If relabelling is non-null (i.e., it points to some NIsomorphism pointer p), then it will be modified to point to a new NIsomorphism that describes the precise relationship between this triangulation and the reconstruction from fromIsoSig(). Specifically, the triangulation that is reconstructed from fromIsoSig() will be combinatorially identical to relabelling.apply(this).

For a full and precise description of the isomorphism signature format, see Simplification paths in the Pachner graphs of closed orientable 3-manifold triangulations, Burton, 2011, arXiv:1110.6080.

Python:
The isomorphism argument is not present. Instead there are two routines: fromIsoSig(), which returns a string only, and fromIsoSigDetail(), which returns a pair (signature, relabelling).
Precondition
If relabelling is non-null, then this triangulation must be non-empty and connected. The facility to return a relabelling for disconnected triangulations may be added to Regina in a later release.
Parameters
relabellingif non-null, this will be modified to point to a new isomorphism describing the relationship between this triangulation and that reconstructed from fromIsoSig(), as described above.
Returns
the isomorphism signature of this triangulation.
See Also
fromIsoSig
bool regina::NTriangulation::isSolidTorus ( ) const

Determines whether this is a triangulation of the solid torus; that is, the unknot complement.

This routine can be used on a triangulation with real boundary triangles, or on an ideal triangulation (in which case all ideal vertices will be assumed to be truncated).

Warning
The algorithms used in this routine rely on normal surface theory and so might be very slow for larger triangulations (although faster tests are used where possible). The routine knowsSolidTorus() 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 either a real (compact) or ideal (non-compact) triangulation of the solid torus.
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::knowsBall ( ) const

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

If this property is indeed already known, future calls to isBall() 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 isBall() and this routine will return true.

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

Warning
This routine does not actually tell you whether this triangulation forms a ball; it merely tells you whether the answer has already been computed (or is very easily computed).
Returns
true if and only if this property is already known or trivial to calculate.
bool regina::NTriangulation::knowsCompressingDisc ( ) const

Is it already known (or trivial to determine) whether or not the underlying 3-manifold contains a compressing disc? See hasCompressingDisc() for further details.

If this property is indeed already known, future calls to hasCompressingDisc() 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 hasCompressingDisc() and this routine will return true.

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

Warning
This routine does not actually tell you whether the underlying 3-manifold has a compressing disc; it merely tells you whether the answer has already been computed (or is very easily computed).
Precondition
This triangulation is valid and is not ideal.
The underlying 3-manifold is irreducible.
Returns
true if and only if this property is already known or trivial to calculate.
bool regina::NTriangulation::knowsHaken ( ) const

Is it already known (or trivial to determine) whether or not the underlying 3-manifold is Haken? See isHaken() for further details.

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

Warning
This routine does not actually tell you whether the underlying 3-manifold is Haken; it merely tells you whether the answer has already been computed (or is very easily computed).
Precondition
This triangulation is valid, closed, orientable and connected.
Returns
true if and only if this property is already known or trivial to calculate.
bool regina::NTriangulation::knowsIrreducible ( ) const

Is it already known (or trivial to determine) whether or not the underlying 3-manifold is irreducible? See isIrreducible() for further details.

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

Warning
This routine does not actually tell you whether the underlying 3-manifold is irreducible; it merely tells you whether the answer has already been computed (or is very easily computed).
Precondition
This triangulation is valid, closed, orientable and connected.
Returns
true if and only if this property is already known or trivial to calculate.
bool regina::NTriangulation::knowsSolidTorus ( ) const

Is it already known (or trivial to determine) whether or not this is a triangulation of a solid torus (that is, the unknot complement)? See isSolidTorus() for further details.

If this property is indeed already known, future calls to isSolidTorus() 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 isSolidTorus() and this routine will return true.

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

Warning
This routine does not actually tell you whether this triangulation forms a solid torus; it merely tells you whether the answer has already been computed (or is very easily computed).
Returns
true if and only if this property is already known or trivial to calculate.
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).

Warning
This routine does not actually tell you whether this triangulation has a splitting surface; it merely tells you whether the answer has already been computed.
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.

Warning
This routine does not actually tell you whether this triangulation forms a 3-sphere; it merely tells you whether the answer has already been computed (or is very easily computed).
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).

Warning
This routine does not actually tell you whether this triangulation is 0-efficient; it merely tells you whether the answer has already been computed.
Returns
true if and only if this property is already known.
NTetrahedron* regina::NTriangulation::layerOn ( NEdge edge)

Performs a layering upon the given boundary edge of the triangulation.

See the NLayering class notes for further details on what a layering entails.

Precondition
The given edge is a boundary edge of this triangulation, and the two boundary triangles on either side of it are distinct.
Parameters
edgethe boundary edge upon which to layer.
Returns
the new tetrahedron provided by the layering.
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 ( std::set< NEdge * > &  edgeSet,
std::set< NVertex * > &  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
edgeSetthe set to be emptied and into which the edges of the maximal forest will be placed.
vertexSetthe set to be emptied and into which the vertices of the maximal forest will be placed.
void regina::NTriangulation::maximalForestInDualSkeleton ( std::set< NTriangle * > &  triangleSet) const

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

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

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

Python:
Not present.
Parameters
triangleSetthe set to be emptied and into which the triangles representing the maximal forest will be placed.
void regina::NTriangulation::maximalForestInSkeleton ( std::set< NEdge * > &  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
edgeSetthe set to be emptied and into which the edges of the maximal forest will be placed.
canJoinBoundariestrue if and only if different boundary components are allowed to be joined by the maximal forest.
void regina::NTriangulation::moveContentsTo ( NTriangulation dest)

Moves the contents of this triangulation into the given destination triangulation, without destroying any pre-existing contents.

That is, all tetrahedra that currently belong to dest will remain there, and all tetrahedra that belong to this triangulation will be moved across as additional tetrahedra in dest.

All NTetrahedron pointers or references will remain valid. After this operation, this triangulation will be empty.

Parameters
destthe triangulation to which tetrahedra should be moved.
NTetrahedron * regina::NTriangulation::newSimplex ( )
inline

A dimension-agnostic alias for newTetrahedron().

This is to assist with writing dimension-agnostic code that can be reused to work in different dimensions.

Here "simplex" refers to a top-dimensional simplex (which for 3-manifold triangulations means a tetrahedron).

See newTetrahedron() for further information.

NTetrahedron * regina::NTriangulation::newSimplex ( const std::string &  desc)
inline

A dimension-agnostic alias for newTetrahedron().

This is to assist with writing dimension-agnostic code that can be reused to work in different dimensions.

Here "simplex" refers to a top-dimensional simplex (which for 3-manifold triangulations means a tetrahedron).

See newTetrahedron() for further information.

NTetrahedron * regina::NTriangulation::newTetrahedron ( )
inline

Creates a new tetrahedron and adds it to this triangulation.

The new tetrahedron will have an empty description. All four faces of the new tetrahedron will be boundary triangles.

The new tetrahedron will become the last tetrahedron in this triangulation.

Returns
the new tetrahedron.
NTetrahedron * regina::NTriangulation::newTetrahedron ( const std::string &  desc)
inline

Creates a new tetrahedron with the given description and adds it to this triangulation.

All four faces of the new tetrahedron will be boundary triangles.

Parameters
descthe description to assign to the new tetrahedron.
Returns
the new tetrahedron.
bool regina::NTriangulation::openBook ( NTriangle t,
bool  check = true,
bool  perform = true 
)

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

This involves taking a triangle meeting the boundary along two edges, and ungluing it to create two new boundary triangles (thus exposing the tetrahedra it initially joined). This move is the inverse of the closeBook() move, and is used to open the way for new shellBoundary() moves.

This move can be done if:

  • the triangle meets the boundary in precisely two edges (and thus also joins two tetrahedra);
  • the vertex between these two edges is a standard boundary vertex (its link is a disc);
  • the remaining edge of the triangle (which is internal to the triangulation) is valid.

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 (triangles, components, etc.) will be reconstructed, which means any pointers to old skeletal objects (such as the argument f) can no longer be used.

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 given triangle is a triangle of this triangulation.
Parameters
tthe triangle about which to perform the move.
checktrue if we are to check whether the move is allowed (defaults to true).
performtrue 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::order ( bool  forceOriented = false)

Relabels tetrahedron vertices in this triangulation to give an ordered triangulation, if possible.

To be an ordered triangulation, all face gluings (when restricted to the tetrahedron face) must be order preserving. In other words, it must be possible to orient all edges of the triangulation in such a fashion that they are consistent with the ordering of the vertices in each tetrahedron.

If it is possible to order this triangulation, the vertices of each tetrahedron will be relabelled accordingly and this routine will return true. Otherwise, this routine will return false and the triangulation will not be changed.

Warning
This routine may be slow, since it backtracks through all possible edge orientations until a consistent one has been found.
Parameters
forceOrientedtrue if the triangulation must be both ordered and oriented, in which case this routine will return false if the triangulation cannot be oriented and ordered at the same time. See orient() for further details.
Returns
true if the triangulation has been successfully ordered as described above, or false if not.
Author
Matthias Goerner
void regina::NTriangulation::orient ( )

Relabels tetrahedron vertices in this triangulation so that all tetrahedra are oriented consistently, if possible.

This routine works by flipping vertices 2 and 3 of each tetrahedron with negative orientation. The result will be a triangulation where the tetrahedron vertices are labelled in a way that preserves orientation across adjacent tetrahedron faces. In particular, every gluing permutation will have negative sign.

If this triangulation includes both orientable and non-orientable components, the orientable components will be oriented as described above and the non-orientable components will be left untouched.

Author
Matthias Goerner
static NTriangulation* regina::NTriangulation::rehydrate ( const std::string &  dehydration)
static

Rehydrates the given alphabetical string into a new triangulation.

See dehydrate() for more information on dehydration strings.

This routine will rehydrate the given string into a new triangulation, and return this new triangulation.

The converse routine dehydrate() can be used to extract a dehydration string from an existing triangulation. Dehydration followed by rehydration might not produce a triangulation identical to the original, but it is guaranteed to produce an isomorphic copy. See dehydrate() for the reasons behind this.

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
dehydrationa dehydrated representation of the triangulation to construct. Case is irrelevant; all letters will be treated as if they were lower case.
Returns
a newly allocated triangulation if the rehydration was successful, or null if the given string could not be rehydrated.
See Also
dehydrate
insertRehydration
void regina::NTriangulation::removeAllSimplices ( )
inline

A dimension-agnostic alias for removeAllTetrahedra().

This is to assist with writing dimension-agnostic code that can be reused to work in different dimensions.

Here "simplex" refers to a top-dimensional simplex (which for 3-manifold triangulations means a tetrahedron).

See removeAllTetrahedra() for further information.

void regina::NTriangulation::removeAllTetrahedra ( )
inline

Removes all tetrahedra from the triangulation.

All tetrahedra will be deallocated.

void regina::NTriangulation::removeSimplex ( NTetrahedron tet)
inline

A dimension-agnostic alias for removeTetrahedron().

This is to assist with writing dimension-agnostic code that can be reused to work in different dimensions.

Here "simplex" refers to a top-dimensional simplex (which for 3-manifold triangulations means a tetrahedron).

See removeTetrahedron() for further information.

void regina::NTriangulation::removeSimplexAt ( unsigned long  index)
inline

A dimension-agnostic alias for removeTetrahedronAt().

This is to assist with writing dimension-agnostic code that can be reused to work in different dimensions.

Here "simplex" refers to a top-dimensional simplex (which for 3-manifold triangulations means a tetrahedron).

See removeTetrahedronAt() for further information.

void 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 be deallocated.

Precondition
The given tetrahedron exists in the triangulation.
Warning
This routine has changed behaviour as of Regina 4.90. In older versions of Regina, the tetrahedron was returned to the user. As of Regina 4.90, the tetrahedron is now destroyed immediately.
Parameters
tetthe tetrahedron to remove.
void 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 be deallocated.

Warning
This routine has changed behaviour as of Regina 4.90. In older versions of Regina, the tetrahedron was returned to the user. As of Regina 4.90, the tetrahedron is now destroyed immediately.
Parameters
indexspecifies which tetrahedron to remove; this should be between 0 and getNumberOfTetrahedra()-1 inclusive.
void regina::NTriangulation::reorderTetrahedraBFS ( bool  reverse = false)

Reorders the tetrahedra of this triangulation using a breadth-first search, so that small-numbered tetrahedra are adjacent to other small-numbered tetrahedra.

Specifically, the reordering will operate as follows. Tetrahedron 0 will remain tetrahedron 0. Its immediate neighbours will be numbered 1, 2, 3 and 4 (though if these neighbours are not distinct then of course fewer labels will be required). Their immediate neighbours will in turn be numbered 5, 6, and so on, ultimately following a breadth-first search throughout the entire triangulation.

If the optional argument reverse is true, then tetrahedron numbers will be assigned in reverse order. That is, tetrahedron 0 will become tetrahedron n-1, its neighbours will become tetrahedra n-2 down to n-5, and so on.

Parameters
reversetrue if the new tetrahedron numbers should be assigned in reverse order, as described above.
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 if:

  • all edges of the tetrahedron are valid;
  • precisely one, two or three faces of the tetrahedron lie in the boundary;
  • if one face lies in the boundary, then the opposite vertex does not lie in the boundary, and no two of the remaining three edges are identified;
  • if two faces lie in the boundary, then the remaining edge does not lie in the boundary, and the remaining two faces of the tetrahedron are not 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 (triangles, components, etc.) will be reconstructed, which means any pointers to old skeletal objects can no longer be used.

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 given tetrahedron is a tetrahedron of this triangulation.
Parameters
tthe tetrahedron upon which to perform the move.
checktrue if we are to check whether the move is allowed (defaults to true).
performtrue 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.
long regina::NTriangulation::simplexIndex ( const NTetrahedron tet) const
inline

A dimension-agnostic alias for tetrahedronIndex().

This is to assist with writing dimension-agnostic code that can be reused to work in different dimensions.

Here "simplex" refers to a top-dimensional simplex (which for 3-manifold triangulations means a tetrahedron).

See tetrahedronIndex() for further information.

void regina::NTriangulation::simplifiedFundamentalGroup ( NGroupPresentation newGroup)
inline

Notifies the triangulation that you have simplified the presentation of its fundamental group.

The old group presentation will be destroyed, and this triangulation will take ownership of the new (hopefully simpler) group that is passed.

This routine is useful for situations in which some external body (such as GAP) has simplified the group presentation better than Regina can.

Regina does not verify that the new group presentation is equivalent to the old, since this is - well, hard.

If the fundamental group has not yet been calculated for this triangulation, this routine will nevertheless take ownership of the new group, under the assumption that you have worked out the group through some other clever means without ever having needed to call getFundamentalGroup() at all.

Note that this routine will not fire a packet change event.

Parameters
newGroupa new (and hopefully simpler) presentation of the fundamental group of this triangulation.
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 moves that do not reduce the number of tetrahedra (such as 4-4 moves or book opening moves) are not used in this routine. Such moves do however feature in intelligentSimplify().

Warning
The specific behaviour of this routine is very likely to change between releases.
Parameters
performtrue 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.
std::string regina::NTriangulation::snapPea ( ) const

Returns a string containing the full contents of a SnapPea data file that describes this triangulation.

This string can, for instance, be used to pass the triangulation to SnapPy without writing to the filesystem.

If you wish to export a triangulation to a SnapPea file, you should use the global function writeSnapPea() instead (which has better performance, and does not require you to construct an enormous intermediate string).

For details on how the SnapPea file will be constructed and what will be included, see the documentation for writeSnapPea().

Precondition
This triangulation is not invalid, and does not contain any boundary triangles.
Returns
a string containing the contents of the corresponding SnapPea data file.
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
componentParentthe packet beneath which the new component triangulations will be inserted, or 0 if they should be inserted directly beneath this triangulation.
setLabelstrue 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.
void regina::NTriangulation::swapContents ( NTriangulation other)

Swaps the contents of this and the given triangulation.

That is, all tetrahedra that belong to this triangulation will be moved to other, and all tetrahedra that belong to other will be moved to this triangulation.

All NTetrahedron pointers or references will remain valid.

Parameters
otherthe triangulation whose contents should be swapped with this.
long regina::NTriangulation::tetrahedronIndex ( 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.

This routine was introduced in Regina 4.5, and replaces the old getTetrahedronIndex(). The name has been changed because, unlike the old routine, it requires that the given tetrahedron belongs to the triangulation (a consequence of some significant memory optimisations).

Precondition
The given tetrahedron is contained in this triangulation.
Warning
Passing a null pointer to this routine will probably crash your program. If you are passing the result of some other routine that might return null (such as NTetrahedron::adjacentTetrahedron), it might be worth explicitly testing for null beforehand.
Parameters
tetspecifies 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.
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 triangle. This can be done iff (i) the edge is valid and non-boundary, and (ii) 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 (triangles, components, etc.) will be reconstructed, which means any pointers to old skeletal objects (such as the argument e) can no longer be used.

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 given edge is an edge of this triangulation.
Parameters
ethe edge about which to perform the move.
checktrue if we are to check whether the move is allowed (defaults to true).
performtrue 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.
long regina::NTriangulation::triangleIndex ( const NTriangle triangle) const
inline

Returns the index of the given triangle in the triangulation.

This routine was introduced in Regina 4.5, and replaces the old getFaceIndex(). The name has been changed because, unlike the old routine, it requires that the given triangle belongs to the triangulation (a consequence of some significant memory optimisations).

Precondition
The given triangle belongs to this triangulation.
Warning
Passing a null pointer to this routine will probably crash your program.
Parameters
trianglespecifies which triangle to find in the triangulation.
Returns
the index of the specified triangle, where 0 is the first triangle, 1 is the second and so on.
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.

These invariants, although computed in the complex field, should all be reals. Thus the return type is an ordinary double.

Precondition
This triangulation is valid, closed and non-empty.
Parameters
rthe integer r as described above; this must be at least 3.
whichRootdetermines 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 valid and non-boundary.
  • The two remaining faces of the tetrahedron are not joined, and the tetrahedron face opposite the given endpoint of the edge is not boundary.
  • Consider the second tetrahedron to be merged (the one joined along the face opposite the given endpoint of the edge). Moreover, consider the two edges of this second tetrahedron 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.

There are additional "distinct and not both boundary" conditions on faces of the second tetrahedron, but those follow automatically from the final condition above.

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 (triangles, components, etc.) will be reconstructed, which means any pointers to old skeletal objects (such as the argument e) can no longer be used.

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 given edge is an edge of this triangulation.
Parameters
ethe edge about which to perform the move.
edgeEndthe 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().
checktrue if we are to check whether the move is allowed (defaults to true).
performtrue 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 ( NTriangle t,
bool  check = true,
bool  perform = true 
)

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

This involves replacing the two tetrahedra joined at that triangle 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 (triangles, components, etc.) will be reconstructed, which means any pointers to old skeletal objects (such as the argument f) can no longer be used.

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 given triangle is a triangle of this triangulation.
Parameters
tthe triangle about which to perform the move.
checktrue if we are to check whether the move is allowed (defaults to true).
performtrue 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 if:

  • the edge is valid and non-boundary;
  • the two tetrahedra are distinct;
  • the edges opposite e in each tetrahedron are distinct and not both boundary;
  • if triangles f1 and f2 from one tetrahedron are to be flattened onto triangles g1 and g2 of the other respectively, then (a) f1 and g1 are distinct, (b) f2 and g2 are distinct, (c) we do not have both f1 = g2 and g1 = f2, (d) we do not have both f1 = f2 and g1 = g2, and (e) we do not have two of the triangles boundary and 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 (triangles, components, etc.) will be reconstructed, which means any pointers to old skeletal objects (such as the argument e) can no longer be used.

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 given edge is an edge of this triangulation.
Parameters
ethe edge about which to perform the move.
checktrue if we are to check whether the move is allowed (defaults to true).
performtrue 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 if:

  • the vertex is non-boundary and has a 2-sphere vertex link;
  • the two tetrahedra are distinct;
  • the triangles opposite v in each tetrahedron are distinct and not both boundary;
  • 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 (triangles, components, etc.) will be reconstructed, which means any pointers to old skeletal objects (such as the argument v) can no longer be used.

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 given vertex is a vertex of this triangulation.
Parameters
vthe vertex about which to perform the move.
checktrue if we are to check whether the move is allowed (defaults to true).
performtrue 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.
long regina::NTriangulation::vertexIndex ( const NVertex vertex) const
inline

Returns the index of the given vertex in the triangulation.

This routine was introduced in Regina 4.5, and replaces the old getVertexIndex(). The name has been changed because, unlike the old routine, it requires that the given vertex belongs to the triangulation (a consequence of some significant memory optimisations).

Precondition
The given vertex belongs to this triangulation.
Warning
Passing a null pointer to this routine will probably crash your program.
Parameters
vertexspecifies 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.
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 provide the user with all the information they could want. The output should be human-readable, should not contain extremely long lines (so users can read the output in a terminal), and should end with a final 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
outthe output stream to which to write.

Reimplemented from regina::ShareableObject.

void regina::NTriangulation::writeTextShort ( std::ostream &  out) const
inlinevirtual

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

The output should be human-readable, should fit on a single line, and should not end with a newline.

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

Implements regina::ShareableObject.

virtual void regina::NTriangulation::writeXMLPacketData ( std::ostream &  out) const
protectedvirtual

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
outthe output stream to which the XML should be written.

Implements regina::NPacket.


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

Copyright © 1999-2013, 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@debian.org).