Regina Calculation Engine
Public Types | Public Member Functions | Static Public Attributes | Protected Types | Protected Member Functions | Protected Attributes | List of all members
regina::Triangulation< 4 > Class Referenceabstract

Represents a 4-dimensional triangulation, typically of a 4-manifold. More...

#include <triangulation/dim4.h>

Inheritance diagram for regina::Triangulation< 4 >:
regina::Packet regina::detail::TriangulationBase< 4 > regina::Output< Packet > regina::SafePointeeBase< Packet > regina::detail::FaceListSuite< dim, dim - 1 > regina::alias::Simplices< TriangulationBase< dim >, dim > regina::alias::SimplexAt< TriangulationBase< dim >, dim, true > regina::alias::FaceOfTriangulation< TriangulationBase< dim >, dim > regina::alias::FacesOfTriangulation< TriangulationBase< dim >, dim > regina::FaceList< dim, subdim >

Public Types

typedef std::vector< Pentachoron< 4 > * >::const_iterator PentachoronIterator
 A dimension-specific alias for SimplexIterator, used to iterate through pentachora. More...
 
typedef FaceList< 4, 3 >::Iterator TetrahedronIterator
 Used to iterate through tetrahedra. More...
 
typedef FaceList< 4, 2 >::Iterator TriangleIterator
 Used to iterate through triangles. More...
 
typedef FaceList< 4, 1 >::Iterator EdgeIterator
 Used to iterate through edges. More...
 
typedef FaceList< 4, 0 >::Iterator VertexIterator
 Used to iterate through vertices. More...
 
typedef Packet SafePointeeType
 The type of object being pointed to. More...
 
typedef std::vector< Simplex< dim > * >::const_iterator SimplexIterator
 Used to iterate through top-dimensional simplices. More...
 
typedef std::vector< Component< dim > * >::const_iterator ComponentIterator
 Used to iterate through connected components. More...
 
typedef std::vector< BoundaryComponent< dim > * >::const_iterator BoundaryComponentIterator
 Used to iterate through boundary components. More...
 

Public Member Functions

std::string str () const
 Returns a short text representation of this object. More...
 
std::string utf8 () const
 Returns a short text representation of this object using unicode characters. More...
 
std::string detail () const
 Returns a detailed text representation of this object. More...
 
bool hasSafePtr () const
 Is there one or more SafePtr currently pointing to this object? More...
 
Iterator begin () const
 Returns an iterator pointing to the first subdim-face. More...
 
Iterator end () const
 Returns an iterator pointing beyond the last subdim-face. More...
 
Constructors and Destructors
 Triangulation ()
 Default constructor. More...
 
 Triangulation (const Triangulation &copy)
 Creates a copy of the given triangulation. More...
 
 Triangulation (const Triangulation &copy, bool cloneProps)
 Creates a new copy of the given triangulation, with the option of whether or not to clone its computed properties also. More...
 
 Triangulation (const std::string &description)
 "Magic" constructor that tries to find some way to interpret the given string as a triangulation. More...
 
virtual ~Triangulation ()
 Destroys this triangulation. More...
 
Packet Administration
virtual void writeTextShort (std::ostream &out) const override
 Writes a short text representation of this object to the given output stream. More...
 
virtual void writeTextLong (std::ostream &out) const override
 Writes a detailed text representation of this object to the given output stream. More...
 
virtual bool dependsOnParent () const override
 Determines if this packet depends upon its parent. More...
 
Pentachora
Pentachoron< 4 > * newPentachoron ()
 A dimension-specific alias for newSimplex(). More...
 
Pentachoron< 4 > * newPentachoron (const std::string &desc)
 A dimension-specific alias for newSimplex(). More...
 
void removePentachoron (Pentachoron< 4 > *tet)
 A dimension-specific alias for removeSimplex(). More...
 
void removePentachoronAt (size_t index)
 A dimension-specific alias for removeSimplexAt(). More...
 
void removeAllPentachora ()
 A dimension-specific alias for removeAllSimplices(). More...
 
Basic Properties
long eulerCharManifold () const
 Returns the Euler characteristic of the corresponding compact manifold. More...
 
bool isIdeal () const
 Determines if this triangulation is ideal. More...
 
bool isClosed () const
 Determines if this triangulation is closed. More...
 
Algebraic Properties
const AbelianGrouphomologyH2 () const
 Returns the second homology group for this triangulation. More...
 
Skeletal Transformations
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 pentachora. More...
 
bool simplifyExhaustive (int height=1, unsigned nThreads=1, ProgressTrackerOpen *tracker=0)
 Attempts to simplify this triangulation using a slow but exhaustive search through the Pachner graph. More...
 
template<typename Action , typename... Args>
bool retriangulate (int height, unsigned nThreads, ProgressTrackerOpen *tracker, Action &&action, Args &&... args) const
 Explores all triangulations that can be reached from this via Pachner moves, without exceeding a given number of additional pentachora. More...
 
bool fiveOneMove (Vertex< 4 > *v, bool check=true, bool perform=true)
 Deprecated function that checks the eligibility of and/or performs a 5-1 Pachner move upon the given vertex. More...
 
bool fourTwoMove (Edge< 4 > *e, bool check=true, bool perform=true)
 Deprecated function that checks the eligibility of and/or performs a 4-2 move about the given edge. More...
 
bool threeThreeMove (Triangle< 4 > *t, bool check=true, bool perform=true)
 Deprecated function that checks the eligibility of and/or performs a 3-3 move about the given triangle. More...
 
bool twoFourMove (Tetrahedron< 4 > *f, bool check=true, bool perform=true)
 Deprecated function that checks the eligibility of and/or performs a 2-4 move about the given facet. More...
 
bool oneFiveMove (Pentachoron< 4 > *p, bool check=true, bool perform=true)
 Deprecated function that checks the eligibility of and/or performs a 1-5 Pachner move upon the given pentachoron. More...
 
bool twoZeroMove (Triangle< 4 > *t, bool check=true, bool perform=true)
 Checks the eligibility of and/or performs a 2-0 move about the given triangle of degree 2. More...
 
bool twoZeroMove (Edge< 4 > *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 (Vertex< 4 > *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 openBook (Tetrahedron< 4 > *t, bool check=true, bool perform=true)
 Checks the eligibility of and/or performs a book opening move about the given tetrahedron. More...
 
bool shellBoundary (Pentachoron< 4 > *p, bool check=true, bool perform=true)
 Checks the eligibility of and/or performs a boundary shelling move on the given pentachoron. More...
 
bool collapseEdge (Edge< 4 > *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...
 
Packet Identification
virtual PacketType type () const =0
 Returns the unique integer ID representing this type of packet. More...
 
virtual std::string typeName () const =0
 Returns an English name for this type of packet. More...
 
const std::string & label () const
 Returns the label associated with this individual packet. More...
 
std::string humanLabel () const
 Returns the label associated with this individual packet, adjusted if necessary for human-readable output. More...
 
std::string adornedLabel (const std::string &adornment) const
 Returns the label of this packet adorned with the given string. More...
 
void setLabel (const std::string &label)
 Sets the label associated with this individual packet. More...
 
std::string fullName () const
 Returns a descriptive text string for the packet. More...
 
Tags
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 > & tags () const
 Returns the set of all tags associated with this packet. More...
 
Event Handling
bool listen (PacketListener *listener)
 Registers the given packet listener to listen for events on this packet. More...
 
bool isListening (PacketListener *listener)
 Determines whether the given packet listener is currently listening for events on this packet. More...
 
bool unlisten (PacketListener *listener)
 Unregisters the given packet listener so that it no longer listens for events on this packet. More...
 
Tree Queries
Packetparent () const
 Determines the parent packet in the tree structure. More...
 
PacketfirstChild () const
 Determines the first child of this packet in the tree structure. More...
 
PacketlastChild () const
 Determines the last child of this packet in the tree structure. More...
 
PacketnextSibling () const
 Determines the next sibling of this packet in the tree structure. More...
 
PacketprevSibling () const
 Determines the previous sibling of this packet in the tree structure. More...
 
Packetroot () const
 Determines the root of the tree to which this packet belongs. More...
 
unsigned levelsDownTo (const Packet *descendant) const
 Counts the number of levels between this packet and its given descendant in the tree structure. More...
 
unsigned levelsUpTo (const Packet *ancestor) const
 Counts the number of levels between this packet and its given ancestor in the tree structure. More...
 
bool isGrandparentOf (const Packet *descendant) const
 Determines if this packet is equal to or an ancestor of the given packet in the tree structure. More...
 
size_t countChildren () const
 Returns the number of immediate children of this packet. More...
 
size_t countDescendants () const
 Returns the total number of strict descendants of this packet. More...
 
size_t totalTreeSize () const
 Determines the total number of packets in the tree or subtree for which this packet is matriarch. More...
 
Tree Manipulation
void insertChildFirst (Packet *child)
 Inserts the given packet as the first child of this packet. More...
 
void insertChildLast (Packet *child)
 Inserts the given packet as the last child of this packet. More...
 
void insertChildAfter (Packet *newChild, Packet *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 (Packet *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 transferChildren (Packet *newParent)
 Cuts all of this packet's children out of the packet tree, and reinserts them as children 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...
 
Searching and Iterating
SubtreeIterator begin ()
 Returns an iterator at the beginning of the range of packets in the subtree rooted at this packet. More...
 
SubtreeIterator end ()
 Returns an iterator beyond the end of the range of packets in the subtree rooted at this packet. More...
 
PacketDescendants descendants () const
 Returns a lightweight object for iterating through all strict descendants of this packet in the packet tree. More...
 
PacketChildren children () const
 Returns a lightweight object for iterating through the immediate children of this packet. More...
 
PacketnextTreePacket ()
 Finds the next packet after this in a complete depth-first iteration of the entire tree structure to which this packet belongs. More...
 
const PacketnextTreePacket () const
 Finds the next packet after this in a complete depth-first iteration of the entire tree structure to which this packet belongs. More...
 
PacketnextTreePacket (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 PacketnextTreePacket (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...
 
PacketfirstTreePacket (const std::string &type)
 Finds the first packet of the requested type in a complete depth-first iteration of the tree structure. More...
 
const PacketfirstTreePacket (const std::string &type) const
 Finds the first packet of the requested type in a complete depth-first iteration of the tree structure. More...
 
PacketfindPacketLabel (const std::string &label)
 Finds the packet with the requested label in the tree or subtree for which this packet is matriarch. More...
 
const PacketfindPacketLabel (const std::string &label) const
 Finds the packet with the requested label in the tree or subtree for which this packet is matriarch. More...
 
Packet Dependencies
bool isPacketEditable () const
 Determines whether this packet can be altered without invalidating or otherwise upsetting any of its immediate children. More...
 
Cloning
Packetclone (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...
 
Simplices
size_t size () const
 Returns the number of top-dimensional simplices in the triangulation. More...
 
const std::vector< Simplex< dim > * > & simplices () const
 Returns all top-dimensional simplices in the triangulation. More...
 
Simplex< dim > * simplex (size_t index)
 Returns the top-dimensional simplex at the given index in the triangulation. More...
 
const Simplex< dim > * simplex (size_t index) const
 Returns the top-dimensional simplex at the given index in the triangulation. More...
 
Simplex< dim > * newSimplex ()
 Creates a new top-dimensional simplex and adds it to this triangulation. More...
 
Simplex< dim > * newSimplex (const std::string &desc)
 Creates a new top-dimensional simplex with the given description and adds it to this triangulation. More...
 
void removeSimplex (Simplex< dim > *simplex)
 Removes the given top-dimensional simplex from this triangulation. More...
 
void removeSimplexAt (size_t index)
 Removes the top-dimensional simplex at the given index in this triangulation. More...
 
void removeAllSimplices ()
 Removes all simplices from the triangulation. More...
 
void swapContents (Triangulation< dim > &other)
 Swaps the contents of this and the given triangulation. More...
 
void moveContentsTo (Triangulation< dim > &dest)
 Moves the contents of this triangulation into the given destination triangulation, without destroying any pre-existing contents. More...
 
Skeletal Queries
size_t countComponents () const
 Returns the number of connected components in this triangulation. More...
 
size_t countBoundaryComponents () const
 Returns the number of boundary components in this triangulation. More...
 
size_t countFaces () const
 Returns the number of subdim-faces in this triangulation. More...
 
std::vector< size_t > fVector () const
 Returns the f-vector of this triangulation, which counts the number of faces of all dimensions. More...
 
const std::vector< Component< dim > * > & components () const
 Returns all connected components of this triangulation. More...
 
const std::vector< BoundaryComponent< dim > * > & boundaryComponents () const
 Returns all boundary components of this triangulation. More...
 
const FaceList< dim, subdim > & faces () const
 Returns an object that allows iteration through and random access to all subdim-faces of this triangulation. More...
 
Component< dim > * component (size_t index) const
 Returns the requested connected component of this triangulation. More...
 
BoundaryComponent< dim > * boundaryComponent (size_t index) const
 Returns the requested boundary component of this triangulation. More...
 
Face< dim, subdim > * face (size_t index) const
 Returns the requested subdim-face of this triangulation. More...
 
Basic Properties
bool isEmpty () const
 Determines whether this triangulation is empty. More...
 
bool isValid () const
 Determines if this triangulation is valid. More...
 
bool hasBoundaryFacets () const
 Determines if this triangulation has any boundary facets. More...
 
size_t countBoundaryFacets () const
 Returns the total number of boundary facets in this triangulation. More...
 
bool isOrientable () const
 Determines if this triangulation is orientable. More...
 
bool isConnected () const
 Determines if this triangulation is connected. More...
 
bool isOriented () const
 Determines if this triangulation is oriented; that is, if the vertices of its top-dimensional simplices are labelled in a way that preserves orientation across adjacent facets. More...
 
long eulerCharTri () const
 Returns the Euler characteristic of this triangulation. More...
 
Algebraic Properties
const GroupPresentationfundamentalGroup () const
 Returns the fundamental group of this triangulation. More...
 
void simplifiedFundamentalGroup (GroupPresentation *newGroup)
 Notifies the triangulation that you have simplified the presentation of its fundamental group. More...
 
const AbelianGrouphomology () const
 Returns the first homology group for this triangulation. More...
 
const AbelianGrouphomologyH1 () const
 Returns the first homology group for this triangulation. More...
 
Skeletal Transformations
void orient ()
 Relabels the vertices of top-dimensional simplices in this triangulation so that all simplices are oriented consistently, if possible. More...
 
void reflect ()
 Relabels the vertices of top-dimensional simplices in this triangulation so that all simplices reflect their orientation. More...
 
bool pachner (Face< dim, k > *f, bool check=true, bool perform=true)
 Checks the eligibility of and/or performs a (dim + 1 - k)-(k + 1) Pachner move about the given k-face. More...
 
Subdivisions, Extensions and Covers
void makeDoubleCover ()
 Converts this triangulation into its double cover. More...
 
void barycentricSubdivision ()
 Does a barycentric subdivision of the triangulation. More...
 
bool finiteToIdeal ()
 Converts each real boundary component into a cusp (i.e., an ideal vertex). More...
 
Decompositions
size_t splitIntoComponents (Packet *componentParent=nullptr, bool setLabels=true)
 Splits a disconnected triangulation into many smaller triangulations, one for each component. More...
 
Isomorphism Testing
bool isIdenticalTo (const Triangulation< dim > &other) const
 Determines if this triangulation is combinatorially identical to the given triangulation. More...
 
std::unique_ptr< Isomorphism< dim > > isIsomorphicTo (const Triangulation< dim > &other) const
 Determines if this triangulation is combinatorially isomorphic to the given triangulation. More...
 
std::unique_ptr< Isomorphism< dim > > isContainedIn (const Triangulation< dim > &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...
 
size_t findAllIsomorphisms (const Triangulation< dim > &other, OutputIterator output) const
 Finds all ways in which this triangulation is combinatorially isomorphic to the given triangulation. More...
 
size_t findAllSubcomplexesIn (const Triangulation< dim > &other, OutputIterator output) 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...
 
bool makeCanonical ()
 Relabel the top-dimensional simplices and their vertices so that this triangulation is in canonical form. More...
 
Building Triangulations
void insertTriangulation (const Triangulation< dim > &source)
 Inserts a copy of the given triangulation into this triangulation. More...
 
void insertConstruction (size_t nSimplices, const int adjacencies[][dim+1], const int gluings[][dim+1][dim+1])
 Inserts a given triangulation into this triangulation, where the given triangulation is described by a pair of integer arrays. More...
 
Exporting Triangulations
std::string isoSig (Isomorphism< dim > **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...
 

Static Public Member Functions

Constructors and Destructors
static void safeDelete (Packet *p)
 Either destroys or orphans the given packet, according to whether it has safe pointers that currently reference it. More...
 

Static Public Attributes

static constexpr int dimension
 A compile-time constant that gives the dimension of the triangulation. More...
 

Protected Types

typedef std::vector< Face< dim, subdim > * >::const_iterator Iterator
 An iterator type for iterating through this list of faces. More...
 

Protected Member Functions

void deleteFaces ()
 Deletes all faces of dimension subdim and below. More...
 
void swapFaces (FaceListSuite< dim, subdim > &other)
 Swaps all faces of dimension subdim and below with those of the given triangulation. More...
 
void fillFVector (std::vector< size_t > &result) const
 Fills the given vector with the first (subdim + 1) elements of the f-vector. More...
 
bool sameFVector (const FaceListSuite< dim, subdim > &other) const
 Tests whether this and the given triangulation have the same number of k-faces, for each facial dimension ksubdim. More...
 
bool sameDegrees (const FaceListSuite< dim, subdim > &other) const
 Tests whether this and the given triangulation have the same k-face degree sequences, for each facial dimension ksubdim. More...
 
bool sameDegrees (const FaceList< dim, subdim > &other) const
 Tests whether this and the given triangulation have the same subdim-face degree sequences. More...
 
Face< dim, subdim > * operator[] (size_t index) const
 Returns the requested subdim-face. More...
 
void push_back (Face< dim, subdim > *face)
 Pushes the given face onto the end of this list. More...
 
void destroy ()
 Destroys all faces in this list, and clears the list itself. More...
 
void swap (FaceList< dim, subdim > &other)
 Swaps all faces in this list with those in the given list. More...
 

Protected Attributes

MarkedVector< Simplex< dim > > simplices_
 The top-dimensional simplices that form the triangulation. More...
 
MarkedVector< BoundaryComponent< dim > > boundaryComponents_
 The components that form the boundary of the triangulation. More...
 
bool valid_
 Is this triangulation valid? See isValid() for details on what this means. More...
 
int topologyLock_
 If non-zero, this will cause Triangulation<dim>::clearAllProperties() to preserve any computed properties that related to the manifold (as opposed to the specific triangulation). More...
 

Subdivisions and Covers

class regina::Face< 4, 4 >
 
class regina::detail::SimplexBase< 4 >
 
class regina::detail::TriangulationBase< 4 >
 
class regina::XMLTriangulationReader< 4 >
 
bool idealToFinite ()
 Converts an ideal triangulation into a finite triangulation. More...
 
static XMLPacketReaderxmlReader (Packet *parent, XMLTreeResolver &resolver)
 
virtual PacketinternalClonePacket (Packet *parent) const override
 Makes a newly allocated copy of this packet. More...
 
virtual void writeXMLPacketData (std::ostream &out) const override
 Writes a chunk of XML containing the data for this packet only. More...
 

File I/O

bool save (const char *filename, bool compressed=true) const
 Saves the subtree rooted at this packet to the given Regina data file, using Regina's native XML file format. More...
 
bool save (std::ostream &s, bool compressed=true) const
 Writes the subtree rooted at this packet to the given output stream, in the format of a Regina XML data file. More...
 
void writeXMLFile (std::ostream &out) const
 Writes the subtree rooted at this packet to the given output stream in Regina's native XML file format. More...
 
std::string internalID () const
 Returns a unique string ID that identifies this packet. More...
 
bool hasOwner () const
 Indicates whether some other object in the calculation engine is responsible for ultimately destroying this object. More...
 
void writeXMLPacketTree (std::ostream &out) const
 Writes a chunk of XML containing the subtree with this packet as matriarch. More...
 

Importing Triangulations

static Triangulation< dim > * 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...
 
void ensureSkeleton () const
 Ensures that all "on demand" skeletal objects have been calculated. More...
 
bool calculatedSkeleton () const
 Determines whether the skeletal objects and properties of this triangulation have been calculated. More...
 
void clearBaseProperties ()
 Clears all properties that are managed by this base class. More...
 
void swapBaseProperties (TriangulationBase< dim > &other)
 Swaps all properties that are managed by this base class, including skeletal data, with the given triangulation. More...
 
void writeXMLBaseProperties (std::ostream &out) const
 Writes a chunk of XML containing properties of this triangulation. More...
 

Detailed Description

Represents a 4-dimensional triangulation, typically of a 4-manifold.

This is a specialisation of the generic Triangulation class template; see the Triangulation documentation for a general overview of how the triangulation classes work.

This 4-dimensional specialisation offers significant extra functionality, including many functions specific to 4-manifolds.

A 4-manifold triangulation is built from pentachora: a pentachoron is a 4-dimensional simplex, with five vertices.

Member Typedef Documentation

◆ BoundaryComponentIterator

typedef std::vector<BoundaryComponent<dim>*>::const_iterator regina::detail::TriangulationBase< dim >::BoundaryComponentIterator
inherited

Used to iterate through boundary components.

◆ ComponentIterator

typedef std::vector<Component<dim>*>::const_iterator regina::detail::TriangulationBase< dim >::ComponentIterator
inherited

Used to iterate through connected components.

◆ SafePointeeType

The type of object being pointed to.

◆ SimplexIterator

typedef std::vector<Simplex<dim>*>::const_iterator regina::detail::TriangulationBase< dim >::SimplexIterator
inherited

Used to iterate through top-dimensional simplices.

Member Function Documentation

◆ barycentricSubdivision()

void regina::detail::TriangulationBase< dim >::barycentricSubdivision
inherited

Does a barycentric subdivision of the triangulation.

This is done in-place, i.e., the triangulation will be modified directly.

Each top-dimensional simplex s is divided into (dim + 1) factorial sub-simplices by placing an extra vertex at the centroid of every face of every dimension. Each of these sub-simplices t is described by a permutation p of (0, ..., dim). The vertices of such a sub-simplex t are:

  • vertex p[0] of s;
  • the centre of edge (p[0], p[1]) of s;
  • the centroid of triangle (p[0], p[1], p[2]) of s;
  • ...
  • the centroid of face (p[0], p[1], p[2], p[dim]) of s, which is the entire simplex s itself.

The sub-simplices have their vertices numbered in a way that mirrors the original simplex s:

  • vertex p[0] of s will be labelled p[0] in t;
  • the centre of edge (p[0], p[1]) of s will be labelled p[1] in t;
  • the centroid of triangle (p[0], p[1], p[2]) of s will be labelled p[2] in t;
  • ...
  • the centroid of s itself will be labelled p[dim] in t.

In particular, if this triangulation is currently oriented, then this barycentric subdivision will preserve the orientation.

If simplex s has index i in the original triangulation, then its sub-simplex corresponding to permutation p will have index ((dim + 1)! * i + p.index()) in the resulting triangulation. In other words: sub-simplices are ordered first according to the original simplex that contains them, and then according to the lexicographical ordering of the corresponding permutations p.

Precondition
dim is one of Regina's standard dimensions. This precondition is a safety net, since in higher dimensions the triangulation would explode too quickly in size (and for the highest dimensions, possibly beyond the limits of size_t).
Warning
In dimensions 3 and 4, both the labelling and ordering of sub-simplices in the subdivided triangulation has changed as of Regina 5.1. (Earlier versions of Regina made no guarantee about the labelling and ordering; these guarantees are also new to Regina 5.1).
Todo:
Lock the topological properties of the underlying manifold, to avoid recomputing them after the subdivision. However, only do this for valid triangulations (since we can have scenarios where invalid triangulations becoming valid and ideal after subdivision, which may change properties such as Triangulation<4>::knownSimpleLinks).

◆ boundaryComponent()

BoundaryComponent< dim > * regina::detail::TriangulationBase< dim >::boundaryComponent ( size_t  index) const
inlineinherited

Returns the requested boundary component of this triangulation.

Note that each time the triangulation changes, all boundary components will be deleted and replaced with new ones. Therefore this object should be considered temporary only.

Parameters
indexthe index of the desired boundary component; this must be between 0 and countBoundaryComponents()-1 inclusive.
Returns
the requested boundary component.

◆ boundaryComponents()

const std::vector< BoundaryComponent< dim > * > & regina::detail::TriangulationBase< dim >::boundaryComponents
inlineinherited

Returns all boundary components of this triangulation.

Note that, in Regina's standard dimensions, each ideal vertex forms its own boundary component, and some invalid vertices do also. See the BoundaryComponent class notes for full details on what constitutes a boundary component in standard and non-standard dimensions.

Bear in mind that each time the triangulation changes, all boundary component objects will be deleted and replaced with new ones. Therefore these boundary component objects should be considered temporary only.

In contrast, this reference to the list of BoundaryComponent objects 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.

◆ calculatedSkeleton()

bool regina::detail::TriangulationBase< dim >::calculatedSkeleton
inlineprotectedinherited

Determines whether the skeletal objects and properties of this triangulation have been calculated.

These are only calculated "on demand", when a skeletal property is first queried.

Returns
true if and only if the skeleton has been calculated.

◆ clearBaseProperties()

void regina::detail::TriangulationBase< dim >::clearBaseProperties ( )
protectedinherited

Clears all properties that are managed by this base class.

This includes deleting all skeletal objects and emptying the corresponding internal lists, as well as clearing other cached properties and deallocating the corresponding memory where required.

Note that TriangulationBase never calls this routine itself. Typically clearBaseProperties() is only ever called by Triangulation<dim>::clearAllProperties(), which in turn is called by "atomic" routines that change the triangluation (before firing packet change events), as well as the Triangulation<dim> destructor.

◆ component()

Component< dim > * regina::detail::TriangulationBase< dim >::component ( size_t  index) const
inlineinherited

Returns the requested connected component of this triangulation.

Note that each time the triangulation changes, all component objects will be deleted and replaced with new ones. Therefore this component object should be considered temporary only.

Parameters
indexthe index of the desired component; this must be between 0 and countComponents()-1 inclusive.
Returns
the requested component.

◆ components()

const std::vector< Component< dim > * > & regina::detail::TriangulationBase< dim >::components
inlineinherited

Returns all connected components of this triangulation.

Note that each time the triangulation changes, all component objects will be deleted and replaced with new ones. Therefore these component objects should be considered temporary only.

In contrast, this reference to the list of all components 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.

◆ countBoundaryComponents()

size_t regina::detail::TriangulationBase< dim >::countBoundaryComponents
inlineinherited

Returns the number of boundary components in this triangulation.

Note that, in Regina's standard dimensions, each ideal vertex forms its own boundary component, and some invalid vertices do also. See the BoundaryComponent class notes for full details on what constitutes a boundary component in standard and non-standard dimensions.

Returns
the number of boundary components.

◆ countBoundaryFacets()

size_t regina::detail::TriangulationBase< dim >::countBoundaryFacets
inlineinherited

Returns the total number of boundary facets in this triangulation.

This routine counts facets of top-dimensional simplices that are not glued to some adjacent top-dimensional simplex.

Returns
the total number of boundary facets.

◆ countComponents()

size_t regina::detail::TriangulationBase< dim >::countComponents
inlineinherited

Returns the number of connected components in this triangulation.

Returns
the number of connected components.

◆ countFaces()

size_t regina::detail::TriangulationBase< dim >::countFaces
inlineinherited

Returns the number of subdim-faces in this triangulation.

Precondition
The template argument subdim is between 0 and dim-1 inclusive.
Python
Python does not support templates. Instead, Python users should call this function in the form countFaces(subdim); that is, the template parameter subdim becomes the first argument of the function.
Returns
the number of subdim-faces.

◆ deleteFaces()

void regina::detail::FaceListSuite< dim >::deleteFaces
inlineprotectedinherited

Deletes all faces of dimension subdim and below.

This routine destroys the corresponding Face objects and clears the lists that contain them.

◆ detail()

std::string regina::Output< Packet , false >::detail ( ) const
inherited

Returns a detailed text representation of this object.

This text may span many lines, and should provide the user with all the information they could want. It should be human-readable, should not contain extremely long lines (which cause problems for users reading the output in a terminal), and should end with a final newline. There are no restrictions on the underlying character set.

Returns
a detailed text representation of this object.

◆ dumpConstruction()

std::string regina::detail::TriangulationBase< dim >::dumpConstruction
inherited

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 gluings between simplices of this trianguation;
  • two additional lines that declare a new Triangulation<dim> 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 simplices. If this triangulation is very large, the returned string will be very large as well.

Returns
the C++ code that was generated.

◆ ensureSkeleton()

void regina::detail::TriangulationBase< dim >::ensureSkeleton
inlineprotectedinherited

Ensures that all "on demand" skeletal objects have been calculated.

◆ eulerCharTri()

long regina::detail::TriangulationBase< dim >::eulerCharTri
inlineinherited

Returns the Euler characteristic of this triangulation.

This will be evaluated strictly as the alternating sum of the number of i-faces (that is, countVertices() - countEdges() + countTriangles() - ...).

Note that this routine handles ideal triangulations in a non-standard way. Since it computes the Euler characteristic of the triangulation (and not the underlying manifold), this routine will treat each ideal boundary component as a single vertex, and not as an entire (dim-1)-dimensional boundary component.

In Regina's standard dimensions, for a routine that handles ideal boundary components properly (by treating them as (dim-1)-dimensional boundary components when computing Euler characteristic), you can use the routine eulerCharManifold() instead.

Returns
the Euler characteristic of this triangulation.

◆ face()

Face< dim, subdim > * regina::detail::TriangulationBase< dim >::face ( size_t  index) const
inlineinherited

Returns the requested subdim-face of this triangulation.

Precondition
The template argument subdim is between 0 and dim-1 inclusive.
Python
Python does not support templates. Instead, Python users should call this function in the form face(subdim, index); that is, the template parameter subdim becomes the first argument of the function.
Parameters
indexthe index of the desired face, ranging from 0 to countFaces<subdim>()-1 inclusive.
Returns
the requested face.

◆ faces()

const FaceList< dim, subdim > & regina::detail::TriangulationBase< dim >::faces
inlineinherited

Returns an object that allows iteration through and random access to all subdim-faces of this triangulation.

Bear in mind that each time the triangulation changes, all face objects will be deleted and replaced with new ones. Therefore these face objects should be considered temporary only.

In contrast, this reference to the FaceList object itself will remain valid and up-to-date for as long as the triangulation exists.

Python
Python users should call this function in the form faces(subdim). It will then return a Python list containing all the subdim-faces of the triangulation. Be warned that, unlike in C++, this Python list will be a snapshot of the faces when this function is called, and will not be kept up-to-date as the triangulation changes.
Returns
access to the list of all subdim-faces.

◆ fillFVector()

void regina::detail::FaceListSuite< dim >::fillFVector ( std::vector< size_t > &  result) const
inlineprotectedinherited

Fills the given vector with the first (subdim + 1) elements of the f-vector.

Specifically, this routine pushes the values f[0], ..., f[subdim] onto the end of the given vector, where f[k] denotes the number of k-faces that this object stores.

Parameters
resultthe vector in which the results will be placed.

◆ findAllIsomorphisms()

size_t regina::detail::TriangulationBase< dim >::findAllIsomorphisms ( const Triangulation< dim > &  other,
OutputIterator  output 
) const
inlineinherited

Finds all ways in which this triangulation is combinatorially isomorphic to the given triangulation.

This routine behaves identically to isIsomorphicTo(), except that instead of returning just one isomorphism, all such isomorphisms are returned.

See the isIsomorphicTo() notes for additional information.

The isomorphisms that are found will be written to the given output iterator. This iterator must accept objects of type Isomorphism<dim>*. As an example, output might be a back_insert_iterator for a std::vector<Isomorphism<dim>*>.

The isomorphisms that are written to the given output iterator will be newly created, and the caller of this routine is responsible for destroying them.

Python
The output argument is not present. Instead, this routine returns a python list containing all of the isomorphisms that were found.
Warning
For large dimensions, this routine can become extremely slow: its running time includes a factor of (dim+1)!.
Parameters
otherthe triangulation to compare with this one.
outputthe output iterator to which the isomorphisms will be written.
Returns
the number of isomorphisms that were found.

◆ findAllSubcomplexesIn()

size_t regina::detail::TriangulationBase< dim >::findAllSubcomplexesIn ( const Triangulation< dim > &  other,
OutputIterator  output 
) const
inlineinherited

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 written to the given output iterator. This iterator must accept objects of type Isomorphism<dim>*. As an example, output might be a back_insert_iterator for a std::vector<Isomorphism<dim>*>.

The isomorphisms that are written to the given output iterator will be newly created, and the caller of this routine is responsible for destroying them.

Warning
For large dimensions, this routine can become extremely slow: its running time includes a factor of (dim+1)!.
Python
Not present.
Parameters
otherthe triangulation in which to search for isomorphic copies of this triangulation.
outputthe output iterator to which the isomorphisms will be written.
Returns
the number of isomorphisms that were found.

◆ finiteToIdeal()

bool regina::detail::TriangulationBase< dim >::finiteToIdeal
inherited

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

Only boundary components formed from real (dim-1)-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 dim-simplex to each boundary (dim-1)-face, and then gluing these new simplices together in a way that mirrors the adjacencies of the underlying boundary facets. Each boundary component will thereby be pushed up through the new simplices and converted into a cusp formed using vertices of these new simplices.

In Regina's standard dimensions, where triangulations also support an idealToFinite() operation, this routine is a loose converse of that operation.

In dimension 2, every boundary component is spherical and so this routine simply fills all the punctures in the underlying surface. (In dimension 2, triangulations cannot have cusps).

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.

◆ fromIsoSig()

static Triangulation<dim>* regina::detail::TriangulationBase< dim >::fromIsoSig ( const std::string &  sig)
staticinherited

Recovers a full triangulation from an isomorphism signature.

See isoSig() for more information on isomorphism signatures. It will be assumed that the signature describes a triangulation of dimension dim.

The triangulation that is returned will be newly created, and it is the responsibility of the caller of this routine to destroy it.

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. In other words, fromIsoSig() may reconstruct the triangulation with its simplices and/or vertices relabelled. The optional argument to isoSig() allows you to determine the precise relabelling that will be used, if you need to know it.

For a full and precise description of the isomorphism signature format for 3-manifold triangulations, see Simplification paths in the Pachner graphs of closed orientable 3-manifold triangulations, Burton, 2011, arXiv:1110.6080. The format for other dimensions is essentially the same, but with minor dimension-specific adjustments.

Warning
Do not mix isomorphism signatures between dimensions! It is possible that the same string could corresponding to both a p-dimensional triangulation and a q-dimensional triangulation for different dimensions p and q.
Parameters
sigthe isomorphism signature of the triangulation to construct. Note that isomorphism signatures are case-sensitive (unlike, for example, dehydration strings for 3-manifolds).
Returns
a newly allocated triangulation if the reconstruction was successful, or null if the given string was not a valid dim-dimensional isomorphism signature.

◆ fundamentalGroup()

const GroupPresentation & regina::detail::TriangulationBase< dim >::fundamentalGroup
inherited

Returns the fundamental group of this triangulation.

The fundamental group is computed in the dual 2-skeleton. This means:

  • If the triangulation contains any ideal vertices, the fundamental group will be calculated as if each such vertex had been truncated.
  • Likewise, if the triangulation contains any invalid faces of dimension 0,1,...,(dim-3), these will effectively be truncated also.
  • In contrast, if the triangulation contains any invalid (dim-2)-faces (i.e., codimension-2-faces that are identified with themselves under a non-trivial map), the fundamental group will be computed without truncating the centroid of the face. For instance, if a 3-manifold triangulation has an edge identified with itself in reverse, then the fundamental group will be computed without truncating the resulting projective plane cusp. This means that, if a barycentric subdivision is performed on a such a triangulation, the result of fundamentalGroup() might 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, fundamentalGroup() should be called again; this will be instantaneous if the group has already been calculated.

Precondition
This triangulation has at most one component.
Warning
In dimension 3, if you are calling this from the subclass SnapPeaTriangulation then any fillings on the cusps will be ignored. (This is the same as for every routine implemented by Regina's Triangulation<3> class.) If you wish to compute the fundamental group with fillings, call SnapPeaTriangulation::fundamentalGroupFilled() instead.
Returns
the fundamental group.

◆ fVector()

std::vector< size_t > regina::detail::TriangulationBase< dim >::fVector
inlineinherited

Returns the f-vector of this triangulation, which counts the number of faces of all dimensions.

The vector that is returned will have length dim+1. If this vector is f, then f[k] will be the number of k-faces for each 0 ≤ kdim.

This routine is significantly more heavyweight than countFaces(). Its advantage is that, unlike the templatised countFaces(), it allows you to count faces whose dimensions are not known until runtime.

Returns
the f-vector of this triangulation.

◆ hasBoundaryFacets()

bool regina::detail::TriangulationBase< dim >::hasBoundaryFacets
inlineinherited

Determines if this triangulation has any boundary facets.

This routine returns true if and only if the triangulation contains some top-dimension simplex with at least one facet that is not glued to an adjacent simplex.

Returns
true if and only if there are boundary facets.

◆ hasSafePtr()

bool regina::SafePointeeBase< Packet >::hasSafePtr
inlineinherited

Is there one or more SafePtr currently pointing to this object?

◆ homology()

const AbelianGroup & regina::detail::TriangulationBase< dim >::homology
inherited

Returns the first homology group for this triangulation.

The homology is computed in the dual 2-skeleton. This means:

  • If the triangulation contains any ideal vertices, the homology will be calculated as if each such vertex had been truncated.
  • Likewise, if the triangulation contains any invalid faces of dimension 0,1,...,(dim-3), these will effectively be truncated also.
  • In contrast, if the triangulation contains any invalid (dim-2)-faces (i.e., codimension-2-faces that are identified with themselves under a non-trivial map), the homology will be computed without truncating the centroid of the face. For instance, if a 3-manifold triangulation has an edge identified with itself in reverse, then the homology will be computed without truncating the resulting projective plane cusp. This means that, if a barycentric subdivision is performed on a such a triangulation, the result of homology() might change.

This routine can also be accessed via the alias homologyH1() (a name that is more specific, but a little longer to type).

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, homology() should be called again; this will be instantaneous if the group has already been calculated.

Warning
In dimension 3, if you are calling this from the subclass SnapPeaTriangulation then any fillings on the cusps will be ignored. (This is the same as for every routine implemented by Regina's Triangulation<3> class.) If you wish to compute homology with fillings, call SnapPeaTriangulation::homologyFilled() instead.
Returns
the first homology group.

◆ homologyH1()

const AbelianGroup & regina::detail::TriangulationBase< dim >::homologyH1
inlineinherited

Returns the first homology group for this triangulation.

This is identical to calling homology(). See the homology() documentation for further details.

Returns
the first homology group.

◆ insertConstruction()

void regina::detail::TriangulationBase< dim >::insertConstruction ( size_t  nSimplices,
const int  adjacencies[][dim+1],
const int  gluings[][dim+1][dim+1] 
)
inherited

Inserts a given triangulation into this triangulation, where the given triangulation is described by a pair of integer arrays.

The main purpose of this routine is to allow users to hard-code triangulations into C++ source files. In particular, all of the simplex gluings can be hard-coded into a pair of integer arrays at the beginning of the source file, avoiding an otherwise tedious sequence of many calls to Simplex<dim>::join(). If you have a particular triangulation that you would like to hard-code in this way, you can call dumpConstruction() to generate the corresponding integer arrays as C++ source code.

This routine will insert an additional nSimplices top-dimensional simplices into this triangulation. We number these simplices 0,1,...,nSimplices-1. The gluings between these new simplices should be stored in the two arrays as follows.

The adjacencies array describes which simplices are joined to which others. Specifically, adjacencies[s][f] indicates which of the new simplices is joined to facet f of simplex s. This should be between 0 and nSimplices-1 inclusive, or -1 if facet f of simplex s is to be left as a boundary facet.

The gluings array describes the particular gluing permutations used to join these simplices together. Specifically, gluings[s][f][0..dim] should describe the permutation used to join facet f of simplex s to its adjacent simplex. These dim+1 integers should be 0,1,...,dim in some order, so that gluings[s][f][i] contains the image of i under this permutation. If facet f of simplex s is to be left as a boundary facet, then gluings[s][f][0..dim] may contain anything (and will be duly ignored).

If this triangulation is empty before this routine is called, then the new simplices will be given indices 0,1,...,nSimplices-1 according to the numbering described above. Otherwise they will be inserted after any pre-existing simplices, and so they will be given larger indices instead. In the latter case, the adjacencies array should still refer to the new simplices as 0,1,...,nSimplices-1, and this routine will handle any renumbering automatically at runtime.

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.

Python
Not present.
Parameters
nSimplicesthe number of additional simplices to insert.
adjacenciesdescribes which simplices are adjace to which others, as described above. This array must have initial dimension at least nSimplices.
gluingsdescribes the specific gluing permutations, as described above. This array must also have initial dimension at least nSimplices.

◆ insertTriangulation()

void regina::detail::TriangulationBase< dim >::insertTriangulation ( const Triangulation< dim > &  source)
inherited

Inserts a copy of the given triangulation into this triangulation.

The top-dimensional simplices of source will be copied into this triangulation in the same order in which they appear in source. That is, if the original size of this triangulation was S, then the simplex at index i in source will be copied into this triangulation as a new simplex at index S+i.

The copies will use the same vertex numbering and descriptions as the original simplices from source, and any gluings between the simplices of source will likewise be copied across as gluings between their copies in this triangulation.

This routine behaves correctly when source is this triangulation.

Parameters
sourcethe triangulation whose copy will be inserted.

◆ isConnected()

bool regina::detail::TriangulationBase< dim >::isConnected
inlineinherited

Determines if this triangulation is connected.

This routine returns false only if there is more than one connected component. In particular, it returns true for the empty triangulation.

Returns
true if and only if this triangulation is connected.

◆ isContainedIn()

std::unique_ptr< Isomorphism< dim > > regina::detail::TriangulationBase< dim >::isContainedIn ( const Triangulation< dim > &  other) const
inlineinherited

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 Isomorphism class notes.

In particular, note that facets of top-dimensional simplices that lie on the boundary of this triangulation need not correspond to boundary facets of other, and that other may contain more top-dimensional simplices 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::unique_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().

Warning
For large dimensions, this routine can become extremely slow: its running time includes a factor of (dim+1)!.
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.

◆ isEmpty()

bool regina::detail::TriangulationBase< dim >::isEmpty
inlineinherited

Determines whether this triangulation is empty.

An empty triangulation is one with no simplices at all.

Returns
true if and only if this triangulation is empty.

◆ isIdenticalTo()

bool regina::detail::TriangulationBase< dim >::isIdenticalTo ( const Triangulation< dim > &  other) const
inherited

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

Here "identical" means that the triangulations have the same number of top-dimensional simplices, with gluings between the same pairs of numbered simplices using the same gluing permutations. In other words, "identical" means that the triangulations are isomorphic via the identity isomorphism.

For the less strict notion of isomorphic triangulations, which allows relabelling of the top-dimensional simplices and their vertices, see isIsomorphicTo() instead.

This test does not examine the textual simplex descriptions, as seen in Simplex<dim>::description(); these may still differ. It also does not test whether lower-dimensional faces are numbered identically (vertices, edges and so on); this routine is only concerned with top-dimensional simplices.

(At the time of writing, two identical triangulations will always number their lower-dimensional faces in the same way. However, it is conceivable that in future versions of Regina there may be situations in which identical triangulations can acquire different numberings for vertices, edges, and so on.)

Parameters
otherthe triangulation to compare with this one.
Returns
true if and only if the two triangulations are combinatorially identical.

◆ isIsomorphicTo()

std::unique_ptr< Isomorphism< dim > > regina::detail::TriangulationBase< dim >::isIsomorphicTo ( const Triangulation< dim > &  other) const
inlineinherited

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

Two triangulations are isomorphic if and only it is possible to relabel their top-dimensional simplices and the (dim+1) vertices of each simplex in a way that makes the two triangulations combinatorially identical, as returned by isIdenticalTo().

Equivalently, two triangulations are isomorphic if and only if there is a one-to-one and onto boundary complete combinatorial isomorphism from this triangulation to other, as described in the Isomorphism class notes.

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

If the triangulations are isomorphic, then this routine returns one such boundary complete isomorphism (i.e., one such relabelling). The isomorphism will be newly constructed, and to assist with memory management, it will be returned as a std::unique_ptr. Thus, to test whether an isomorphism exists without having to explicitly manage with the isomorphism itself, you can just call if (isIsomorphicTo(other).get()), in which case the newly created isomorphism (if it exists) will be automatically destroyed.

There may be many such isomorphisms between the two triangulations. If you need to find all such isomorphisms, you may call findAllIsomorphisms() instead.

If you need to ensure that top-dimensional simplices are labelled the same in both triangulations (i.e., that the triangulations are related by the identity isomorphism), you should call the stricter test isIdenticalTo() instead.

Warning
For large dimensions, this routine can become extremely slow: its running time includes a factor of (dim+1)!.
Todo:
Optimise: Improve the complexity by choosing a simplex mapping from each component and following gluings to determine the others.
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.

◆ isOrientable()

bool regina::detail::TriangulationBase< dim >::isOrientable
inlineinherited

Determines if this triangulation is orientable.

Returns
true if and only if this triangulation is orientable.

◆ isOriented()

bool regina::detail::TriangulationBase< dim >::isOriented
inherited

Determines if this triangulation is oriented; that is, if the vertices of its top-dimensional simplices are labelled in a way that preserves orientation across adjacent facets.

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 top-dimensional simplices to be oriented consistently as described above.

A non-orientable triangulation can never be oriented.

Returns
true if and only if all top-dimensional simplices are oriented consistently.
Author
Matthias Goerner

◆ isoSig()

std::string regina::detail::TriangulationBase< dim >::isoSig ( Isomorphism< dim > **  relabelling = 0) const
inherited

Constructs the isomorphism signature for this triangulation.

An isomorphism signature is a compact text representation of a triangulation that uniquely determines the triangulation up to combinatorial isomorphism. That is, two triangulations of dimension dim 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 top-dimenisonal simplices.

Whilst the format of an isomorphism signature bears some similarity to dehydration strings for 3-manifolds, they are more general: isomorphism signatures can be used with any triangulations, including closed, bounded and/or disconnected triangulations, as well as triangulations with many simplices. Note also that 3-manifold dehydration strings are not unique up to isomorphism (they depend on the particular labelling of tetrahedra).

The time required to construct the isomorphism signature of a triangulation is O((dim!) n^2 log^2 n). Whilst this is fine for large triangulation, it will be extremly slow for large dimensions.

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 Isomorphism pointer p), then it will be modified to point to a new isomorphism 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 for 3-manifold triangulations, see Simplification paths in the Pachner graphs of closed orientable 3-manifold triangulations, Burton, 2011, arXiv:1110.6080. The format for other dimensions is essentially the same, but with minor dimension-specific adjustments.

Python
The isomorphism argument is not present. Instead there are two routines: isoSig(), which returns a string only, and isoSigDetail(), 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.
Warning
Do not mix isomorphism signatures between dimensions! It is possible that the same string could corresponding to both a p-dimensional triangulation and a q-dimensional triangulation for different dimensions p and q.
Parameters
relabellingif this is non-null, it will be modified to point to a new isomorphism that describes the relationship between this triangulation and the triangulation that will be reconstructed from fromIsoSig(), as described above.
Returns
the isomorphism signature of this triangulation.

◆ isoSigComponentSize()

static size_t regina::detail::TriangulationBase< dim >::isoSigComponentSize ( const std::string &  sig)
staticinherited

Deduces the number of top-dimensional simplices in a connected triangulation from its isomorphism signature.

See isoSig() for more information on isomorphism signatures. It will be assumed that the signature describes a triangulation of dimension dim.

If the signature describes a connected triangulation, this routine will simply return the size of that triangulation (e.g., the number of tetrahedra in the case dim = 3). You can also pass an isomorphism signature that describes a disconnected triangulation; however, this routine will only return the number of top-dimensional simplices in the first connected component. If you need the total size of a disconnected triangulation, you will need to reconstruct the full triangulation by calling fromIsoSig() instead.

This routine is very fast, since it only examines the first few characters of the isomorphism signature (in which the size of the first component is encoded). However, a side-effect of this is that it is possible to pass an invalid isomorphism signature and still receive a positive result. If you need to test whether a signature is valid or not, you must call fromIsoSig() instead, which will examine the entire signature in full.

Warning
Do not mix isomorphism signatures between dimensions! It is possible that the same string could corresponding to both a p-dimensional triangulation and a q-dimensional triangulation for different dimensions p and q.
Parameters
sigthe isomorphism signature of a dim-dimensional triangulation. Note that isomorphism signature are case-sensitive (unlike, for example, dehydration strings for 3-manifolds).
Returns
the number of top-dimensional simplices in the first connected component, or 0 if this could not be determined because the given string was not a valid isomorphism signature.

◆ isValid()

bool regina::detail::TriangulationBase< dim >::isValid
inlineinherited

Determines if this triangulation is valid.

There are several conditions that might make a dim-dimensional triangulation invalid:

  1. if some face is identified with itself under a non-identity permutation (e.g., an edge is identified with itself in reverse, or a triangle is identified with itself under a rotation);
  2. if some subdim-face does not have an appropriate link. Here the meaning of "appropriate" depends upon the type of face:
    • for a face that belongs to some boundary facet(s) of this triangulation, its link must be a topological ball;
    • for a vertex that does not belong to any boundary facets, its link must be a closed (dim - 1)-manifold;
    • for a (subdim ≥ 1)-face that does not belong to any boundary facets, its link must be a topological sphere.

Condition (1) is tested for all dimensions dim. Condition (2) is more difficult, since it relies on undecidable problems. As a result, (2) is only tested when dim is one of Regina's standard dimensions.

If a triangulation is invalid then you can call Face<dim, subdim>::isValid() to discover exactly which face(s) are responsible, and you can call Face<dim, subdim>::hasBadIdentification() and/or Face<dim, subdim>::hasBadLink() to discover exactly which conditions fail.

Note that all invalid vertices are considered to be on the boundary; see isBoundary() for details.

Returns
true if and only if this triangulation is valid.

◆ makeCanonical()

bool regina::detail::TriangulationBase< dim >::makeCanonical ( )
inherited

Relabel the top-dimensional simplices and their vertices so that this triangulation is in canonical form.

This is essentially the lexicographically smallest labelling when the facet gluings are written out in order.

Two triangulations are isomorphic if and only if their canonical forms are identical.

The lexicographic ordering assumes that the facet gluings are written in order of simplex index and then facet number. Each gluing is written as the destination simplex index followed by the gluing permutation (which in turn is written as the images of 0,1,...,dim in order).

Precondition
This routine currently works only when the triangulation is connected. It may be extended to work with disconnected triangulations in later versions of Regina.
Returns
true if the triangulation was changed, or false if the triangulation was in canonical form to begin with.

◆ makeDoubleCover()

void regina::detail::TriangulationBase< dim >::makeDoubleCover
inherited

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.

◆ moveContentsTo()

void regina::detail::TriangulationBase< dim >::moveContentsTo ( Triangulation< dim > &  dest)
inherited

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

All top-dimensional simplices that currently belong to dest will remain there (and will keep the same indices in dest). All top-dimensional simplices that belong to this triangulation will be moved into dest also (but in general their indices will change).

This triangulation will become empty as a result.

Any pointers or references to Simplex<dim> objects will remain valid.

Precondition
dest is not this triangulation.
Parameters
destthe triangulation into which simplices should be moved.

◆ newSimplex() [1/2]

Simplex< dim > * regina::detail::TriangulationBase< dim >::newSimplex
inherited

Creates a new top-dimensional simplex and adds it to this triangulation.

The new simplex will have an empty description. All (dim+1) facets of the new simplex will be boundary facets.

The new simplex will become the last simplex in this triangulation; that is, it will have index size()-1.

Returns
the new simplex.

◆ newSimplex() [2/2]

Simplex< dim > * regina::detail::TriangulationBase< dim >::newSimplex ( const std::string &  desc)
inherited

Creates a new top-dimensional simplex with the given description and adds it to this triangulation.

All (dim+1) facets of the new simplex will be boundary facets.

Descriptions are optional, may have any format, and may be empty. How descriptions are used is entirely up to the user.

The new simplex will become the last simplex in this triangulation; that is, it will have index size()-1.

Parameters
descthe description to give to the new simplex.
Returns
the new simplex.

◆ orient()

void regina::detail::TriangulationBase< dim >::orient
inherited

Relabels the vertices of top-dimensional simplices in this triangulation so that all simplices are oriented consistently, if possible.

This routine works by flipping vertices (dim - 1) and dim of each top-dimensional simplices that has negative orientation. The result will be a triangulation where the top-dimensional simplices have their vertices labelled in a way that preserves orientation across adjacent facets. 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.

◆ pachner()

bool regina::detail::TriangulationBase< dim >::pachner ( Face< dim, k > *  f,
bool  check = true,
bool  perform = true 
)
inlineinherited

Checks the eligibility of and/or performs a (dim + 1 - k)-(k + 1) Pachner move about the given k-face.

This involves replacing the (dim + 1 - k) top-dimensional simplices meeting that k-face with (k + 1) new top-dimensional simplices joined along a new internal (dim - k)-face. This can be done iff (i) the given k-face is valid and non-boundary; (ii) the (dim + 1 - k) top-dimensional simplices that contain it are distinct; and (iii) these simplices are joined in such a way that the link of the given k-face is the standard triangulation of the (dim - 1 - k)-sphere as the boundary of a (dim - k)-simplex.

If the routine is asked to both check and perform, the move will only be performed if the check shows it is legal. In In the special case k = dim, the move is always legal and so the check argument will simply be ignored.

Note that after performing this move, all skeletal objects (facets, components, etc.) will be reconstructed, which means any pointers to old skeletal objects (such as the argument v) can no longer be used.

If this triangulation is currently oriented, then this Pachner move will label the new top-dimensional simplices in a way that preserves the orientation.

See the page on Pachner moves on triangulations for definitions and terminology relating to Pachner moves. After the move, the new belt face will be formed from vertices 0,1,...,(dim - k) of simplices().back().

Warning
For the case k = dim in Regina's standard dimensions, the labelling of the belt face has changed as of Regina 5.96 (the first prerelease for Regina 6.0). In versions 5.1 and earlier, the belt face was simplices().back()->vertex(dim), and as of version 5.96 it is now simplices().back()->vertex(0). The deprecated routines Triangulation<2>::oneThreeMove(), Triangulation<3>::oneFourMove() and Triangulation<4>::oneFiveMove() maintain the old behaviour if you need it.
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 k-face is a k-face of this triangulation.
Parameters
fthe k-face 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.
Template Parameters
kthe dimension of the given face. This must be between 0 and (dim) inclusive. You can still perform a Pachner move about a 0-face dim-face, but these moves use specialised implementations (as opposed to this generic template implementation).

◆ reflect()

void regina::detail::TriangulationBase< dim >::reflect
inherited

Relabels the vertices of top-dimensional simplices in this triangulation so that all simplices reflect their orientation.

In particular, if this triangulation is oriented, then it will be converted into an isomorphic triangulation with the opposite orientation.

This routine works by flipping vertices (dim - 1) and dim of every top-dimensional simplex.

◆ removeAllSimplices()

void regina::detail::TriangulationBase< dim >::removeAllSimplices
inlineinherited

Removes all simplices from the triangulation.

As a result, this triangulation will become empty.

All of the simplices that belong to this triangulation will be destroyed immediately.

◆ removeSimplex()

void regina::detail::TriangulationBase< dim >::removeSimplex ( Simplex< dim > *  simplex)
inlineinherited

Removes the given top-dimensional simplex from this triangulation.

The given simplex will be unglued from any adjacent simplices (if any), and will be destroyed immediately.

Precondition
The given simplex is a top-dimensional simplex in this triangulation.
Parameters
simplexthe simplex to remove.

◆ removeSimplexAt()

void regina::detail::TriangulationBase< dim >::removeSimplexAt ( size_t  index)
inlineinherited

Removes the top-dimensional simplex at the given index in this triangulation.

This is equivalent to calling removeSimplex(simplex(index)).

The given simplex will be unglued from any adjacent simplices (if any), and will be destroyed immediately.

Parameters
indexspecifies which top-dimensionalsimplex to remove; this must be between 0 and size()-1 inclusive.

◆ sameDegrees()

bool regina::detail::FaceListSuite< dim >::sameDegrees ( const FaceListSuite< dim, subdim > &  other) const
inlineprotectedinherited

Tests whether this and the given triangulation have the same k-face degree sequences, for each facial dimension ksubdim.

For the purposes of this routine, degree sequences are considered to be unordered.

Precondition
This and the given triangulation are known to have the same number of k-faces as each other, for each facial dimension ksubdim.
Parameters
otherthe triangulation to compare against this.
Returns
true if and only if all degree sequences considered are equal.

◆ sameFVector()

bool regina::detail::FaceListSuite< dim >::sameFVector ( const FaceListSuite< dim, subdim > &  other) const
inlineprotectedinherited

Tests whether this and the given triangulation have the same number of k-faces, for each facial dimension ksubdim.

Parameters
otherthe triangulation to compare against this.
Returns
true if and only if the face counts considered are identical for both triangluations.

◆ simplex() [1/2]

Simplex< dim > * regina::detail::TriangulationBase< dim >::simplex ( size_t  index)
inlineinherited

Returns the top-dimensional simplex at the given index in the triangulation.

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

Parameters
indexspecifies which simplex to return; this value should be between 0 and size()-1 inclusive.
Returns
the indexth top-dimensional simplex.

◆ simplex() [2/2]

const Simplex< dim > * regina::detail::TriangulationBase< dim >::simplex ( size_t  index) const
inlineinherited

Returns the top-dimensional simplex at the given index in the triangulation.

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

Parameters
indexspecifies which simplex to return; this value should be between 0 and size()-1 inclusive.
Returns
the indexth top-dimensional simplex.

◆ simplices()

const std::vector< Simplex< dim > * > & regina::detail::TriangulationBase< dim >::simplices
inlineinherited

Returns all top-dimensional simplices in the triangulation.

The reference that is returned will remain valid for as long as the triangulation exists: even as simplices are added and/or removed, it will always reflect the simplices that are currently in the triangulation.

Python
This routine returns a python list.
Returns
the list of all top-dimensional simplices.

◆ simplifiedFundamentalGroup()

void regina::detail::TriangulationBase< dim >::simplifiedFundamentalGroup ( GroupPresentation newGroup)
inlineinherited

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 fundamentalGroup() 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.

◆ size()

size_t regina::detail::TriangulationBase< dim >::size
inlineinherited

Returns the number of top-dimensional simplices in the triangulation.

Returns
The number of top-dimensional simplices.

◆ splitIntoComponents()

size_t regina::detail::TriangulationBase< dim >::splitIntoComponents ( Packet componentParent = nullptr,
bool  setLabels = true 
)
inherited

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 (i.e., this triangulation) will be left unchanged.

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

By default, this routine will assign sensible packet labels to each of the new component triangulations. If these component triangulations are only temporary objects used as part of some larger algorithm, then labels are unnecessary - in this case you can pass setLabels as false to avoid the (small) overhead that these packet labels incur.

Parameters
componentParentthe packet beneath which the new component triangulations will be inserted, or null if they should be inserted directly beneath this triangulation.
setLabelstrue if the new component triangulations should be assigned sensible packet labels, or false if they should be left without labels at all.
Returns
the number of new component triangulations constructed.

◆ str()

std::string regina::Output< Packet , false >::str ( ) const
inherited

Returns a short text representation of this object.

This text should be human-readable, should fit on a single line, and should not end with a newline. Where possible, it should use plain ASCII characters.

Python
In addition to str(), this is also used as the Python "stringification" function str().
Returns
a short text representation of this object.

◆ swapBaseProperties()

void regina::detail::TriangulationBase< dim >::swapBaseProperties ( TriangulationBase< dim > &  other)
protectedinherited

Swaps all properties that are managed by this base class, including skeletal data, with the given triangulation.

Note that TriangulationBase never calls this routine itself. Typically swapBaseProperties() is only ever called by Triangulation<dim>::swapAllProperties(), which in turn is called by swapContents().

Parameters
otherthe triangulation whose properties should be swapped with this.

◆ swapContents()

void regina::detail::TriangulationBase< dim >::swapContents ( Triangulation< dim > &  other)
inherited

Swaps the contents of this and the given triangulation.

All top-dimensional simplices that belong to this triangulation will be moved to other, and all top-dimensional simplices that belong to other will be moved to this triangulation. Likewise, all skeletal objects (such as lower-dimensional faces, components, and boundary components) and all cached properties (such as homology and fundamental group) will be swapped.

In particular, any pointers or references to Simplex<dim> and/or Face<dim, subdim> objects will remain valid.

This routine will behave correctly if other is in fact this triangulation.

Parameters
otherthe triangulation whose contents should be swapped with this.

◆ swapFaces()

void regina::detail::FaceListSuite< dim >::swapFaces ( FaceListSuite< dim, subdim > &  other)
inlineprotectedinherited

Swaps all faces of dimension subdim and below with those of the given triangulation.

Parameters
otherthe face storage for the triangulation whose faces are to be swapped with this.

◆ utf8()

std::string regina::Output< Packet , false >::utf8 ( ) const
inherited

Returns a short text representation of this object using unicode characters.

Like str(), this text should be human-readable, should fit on a single line, and should not end with a newline. In addition, it may use unicode characters to make the output more pleasant to read. This string will be encoded in UTF-8.

Returns
a short text representation of this object.

◆ writeXMLBaseProperties()

void regina::detail::TriangulationBase< dim >::writeXMLBaseProperties ( std::ostream &  out) const
protectedinherited

Writes a chunk of XML containing properties of this triangulation.

This routine covers those properties that are managed by this base class TriangulationBase and that have already been computed for this triangulation.

This routine is typically called from within Triangulation<dim>::writeXMLPacketData(). The XML elements that it writes are child elements of the packet element.

Parameters
outthe output stream to which the XML should be written.

Member Data Documentation

◆ boundaryComponents_

MarkedVector<BoundaryComponent<dim> > regina::detail::TriangulationBase< dim >::boundaryComponents_
protectedinherited

The components that form the boundary of the triangulation.

◆ dimension

constexpr int regina::detail::TriangulationBase< dim >::dimension
staticconstexprinherited

A compile-time constant that gives the dimension of the triangulation.

◆ simplices_

MarkedVector<Simplex<dim> > regina::detail::TriangulationBase< dim >::simplices_
protectedinherited

The top-dimensional simplices that form the triangulation.

◆ topologyLock_

int regina::detail::TriangulationBase< dim >::topologyLock_
protectedinherited

If non-zero, this will cause Triangulation<dim>::clearAllProperties() to preserve any computed properties that related to the manifold (as opposed to the specific triangulation).

This allows you to avoid recomputing expensive invariants when the underlying manifold is retriangulated.

This property should be managed by creating and destroying TopologyLock objects. The precise value of topologyLock_ indicates the number of TopologyLock objects that currently exist for this triangulation.

◆ valid_

bool regina::detail::TriangulationBase< dim >::valid_
protectedinherited

Is this triangulation valid? See isValid() for details on what this means.


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

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