Regina Calculation Engine

Represents a saturated block in a Seifert fibred space. More...
#include <subcomplex/nsatblock.h>
Public Types  
typedef std::set< NTetrahedron * >  TetList 
The data structure used to store a list of tetrahedra that should not be examined by isBlock(). More...  
Public Member Functions  
NSatBlock (const NSatBlock &cloneMe)  
Creates a new clone of the given block. More...  
~NSatBlock ()  
Destroys all internal arrays. More...  
virtual NSatBlock *  clone () const =0 
Returns a newly created clone of this saturated block structure. More...  
unsigned  nAnnuli () const 
Returns the number of annuli on the boundary of this saturated block. More...  
const NSatAnnulus &  annulus (unsigned which) const 
Returns details of the requested annulus on the boundary of this saturated block. More...  
bool  twistedBoundary () const 
Is the ring of boundary annuli twisted to form a long Mobius strip? More...  
bool  hasAdjacentBlock (unsigned whichAnnulus) const 
Returns whether there is another saturated block listed as being adjacent to the given boundary annulus of this block. More...  
NSatBlock *  adjacentBlock (unsigned whichAnnulus) const 
Returns the saturated block listed as being adjacent to the given boundary annulus of this block. More...  
unsigned  adjacentAnnulus (unsigned whichAnnulus) const 
Returns which specific annulus of the adjacent block is listed as being adjacent to the given boundary annulus of this block. More...  
bool  adjacentReflected (unsigned whichAnnulus) const 
Returns whether the adjacency along the given boundary annulus of this block is reflected. More...  
bool  adjacentBackwards (unsigned whichAnnulus) const 
Returns whether the adjacency along the given boundary annulus of this block is backwards. More...  
void  setAdjacent (unsigned whichAnnulus, NSatBlock *adjBlock, unsigned adjAnnulus, bool adjReflected, bool adjBackwards) 
Lists the given saturated block as being adjacent to the given boundary annulus of this block. More...  
virtual void  adjustSFS (NSFSpace &sfs, bool reflect) const =0 
Adjusts the given Seifert fibred space to insert the contents of this saturated block. More...  
virtual void  transform (const NTriangulation *originalTri, const NIsomorphism *iso, NTriangulation *newTri) 
Adjusts the structure of this block according to the given isomorphism between triangulations. More...  
void  nextBoundaryAnnulus (unsigned thisAnnulus, NSatBlock *&nextBlock, unsigned &nextAnnulus, bool &refVert, bool &refHoriz, bool followPrev) 
Finds the next (or previous) boundary annulus around from this, treating all adjacent blocks as part of a single large saturated region. More...  
std::string  getAbbr (bool tex=false) const 
Returns an abbreviated name or symbol for this block. More...  
virtual void  writeAbbr (std::ostream &out, bool tex=false) const =0 
Writes an abbreviated name or symbol for this block to the given output stream. More...  
bool  operator< (const NSatBlock &compare) const 
Implements a consistent ordering of saturated blocks. More...  
Public Member Functions inherited from regina::ShareableObject  
ShareableObject ()  
Default constructor that does nothing. More...  
virtual  ~ShareableObject () 
Default destructor that does nothing. More...  
virtual void  writeTextShort (std::ostream &out) const =0 
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...  
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 NSatBlock *  isBlock (const NSatAnnulus &annulus, TetList &avoidTets) 
Determines whether the given annulus is in fact a boundary annulus for a recognised type of saturated block. More...  
Protected Member Functions  
NSatBlock (unsigned nAnnuli, bool twistedBoundary=false)  
Constructor for a block with the given number of annuli on the boundary. More...  
Static Protected Member Functions  
static bool  isBad (NTetrahedron *t, const TetList &list) 
Determines whether the given tetrahedron is contained within the given list. More...  
template<class List >  
static bool  isBad (NTetrahedron *t, const List &list) 
Determines whether the given tetrahedron is contained within the given list. More...  
static bool  notUnique (NTetrahedron *test) 
Determines whether the given tetrahedron pointer is null. More...  
static bool  notUnique (NTetrahedron *test, NTetrahedron *other1) 
Determines whether the given tetrahedron pointer is null or equal to another from the given list. More...  
static bool  notUnique (NTetrahedron *test, NTetrahedron *other1, NTetrahedron *other2) 
Determines whether the given tetrahedron pointer is null or equal to another from the given list. More...  
static bool  notUnique (NTetrahedron *test, NTetrahedron *other1, NTetrahedron *other2, NTetrahedron *other3) 
Determines whether the given tetrahedron pointer is null or equal to another from the given list. More...  
static bool  notUnique (NTetrahedron *test, NTetrahedron *other1, NTetrahedron *other2, NTetrahedron *other3, NTetrahedron *other4) 
Determines whether the given tetrahedron pointer is null or equal to another from the given list. More...  
Protected Attributes  
unsigned  nAnnuli_ 
The number of boundary annuli. More...  
NSatAnnulus *  annulus_ 
Details of each boundary annulus, as seen from the inside of this saturated block. More...  
bool  twistedBoundary_ 
Is the ring of boundary annuli twisted to form a Mobius band? More...  
NSatBlock **  adjBlock_ 
The saturated block joined to each boundary annulus; this may be null if there is no adjacency or if this information is not known. More...  
unsigned *  adjAnnulus_ 
Describes which specific annulus of the adjacent saturated block is joined to each boundary annulus of this block. More...  
bool *  adjReflected_ 
Describes whether the adjacency for each boundary annulus is reflected (see the class notes above). More...  
bool *  adjBackwards_ 
Describes whether the adjacency for each boundary annulus is backwards (see the class notes above). More...  
Represents a saturated block in a Seifert fibred space.
A saturated block is a connected set of tetrahedra built from a subset of fibres (no fibres may enter or exit the boundary of the block). In addition, the boundary of this block must be a ring of saturated annuli, as described by the NSatAnnulus class. Aside from this ring of saturated annuli, there may be no other boundary triangles within the block.
The boundary annuli are numbered consecutively as illustrated below, where the markings 0 and 1 within the triangles represent the first and second triangle of each annulus (see the NSatAnnulus class notes for details). Note that the following diagram is viewed from inside the block.
+++++++ 0 /0 /0 /0 /0 /0 / ...  /  /  /  /  /  /  ... / 1/ 1/ 1/ 1/ 1/ 1 +++++++ Annulus # ... n2 n1 0 1 2 3 ...
The ring of boundary annuli may optionally be twisted, so that together the annuli form a long Mobius band. In this case, for the purposes of labelling and marking annuli, the twist occurs between annuli n1 and 0. Be careful when dealing with blocks with twisted boundaries, since with twists it is possible to identify an edge with itself in reverse (thus producing something that is not a 3manifold triangulation).
Each saturated block corresponds to a piece of the base orbifold of the larger Seifert fibred space. For the purpose of connecting the base orbifold together, we assume that the boundary of this particular piece runs horizontally in the diagram above (specifically following the horizontal edges of the boundary annuli, as described in the NSatAnnulus class notes). Insisting on such a boundary may lead to (1,k) twists within the block; these are accounted for by the virtual adjustSFS() routine.
Saturated blocks are generally joined to one another (or themselves) along their boundary annuli. For this purpose, each saturated block contains a list of which annulus of this block is adjacent to which annulus of which other block. Adjacencies may be reflected, meaning that the adjacent annulus has its fibres reversed (i.e., the adjacent annulus has undergone an uptodown reflection); they may also be backwards, meaning that the first triangle of one annulus is joined to the second triangle of the other (and vice versa).
typedef std::set<NTetrahedron*> regina::NSatBlock::TetList 
The data structure used to store a list of tetrahedra that should not be examined by isBlock().
regina::NSatBlock::NSatBlock  (  const NSatBlock &  cloneMe  ) 
Creates a new clone of the given block.
Note that the new adjBlock_ array will contain pointers to the same adjacent blocks as the original. That is, adjacent blocks will not be cloned also; instead pointers to adjacent blocks will simply be copied across.
cloneMe  the saturated block to clone. 

inline 
Destroys all internal arrays.
Note that any adjacent blocks that are referenced by the adjBlock array will not be destroyed.

inlineprotected 
Constructor for a block with the given number of annuli on the boundary.
All arrays will be constructed but their contents will remain uninitialised, with the exception that the adjBlock array will be filled with null pointers.
nAnnuli  the number of annuli on the boundary of this block; this must be strictly positive. 
twistedBoundary  true if the ring of boundary annuli is twisted to form a long Mobius band, or false (the default) if it is not. 

inline 
Returns which specific annulus of the adjacent block is listed as being adjacent to the given boundary annulus of this block.
whichAnnulus  indicates which boundary annulus of this block should be examined; this must be between 0 and nAnnuli()1 inclusive. 

inline 
Returns whether the adjacency along the given boundary annulus of this block is backwards.
See the class notes for a discussion of backwards adjacencies.
whichAnnulus  indicates which boundary annulus of this block should be examined; this must be between 0 and nAnnuli()1 inclusive. 
true
if the corresponding adjacency is backwards, or false
if it is not.

inline 
Returns the saturated block listed as being adjacent to the given boundary annulus of this block.
whichAnnulus  indicates which boundary annulus of this block should be examined; this must be between 0 and nAnnuli()1 inclusive. 

inline 
Returns whether the adjacency along the given boundary annulus of this block is reflected.
See the class notes for a discussion of reflected adjacencies.
whichAnnulus  indicates which boundary annulus of this block should be examined; this must be between 0 and nAnnuli()1 inclusive. 
true
if the corresponding adjacency is reflected, or false
if it is not.

pure virtual 
Adjusts the given Seifert fibred space to insert the contents of this saturated block.
In particular, the space should be adjusted as though an ordinary solid torus (base orbifold a disc, no twists or exceptional fibres) had been replaced by this block. This description does not make sense for blocks with twisted boundary; the twisted case is discussed below.
If the argument reflect is true
, it should be assumed that this saturated block is being reflected before being inserted into the larger Seifert fibred space. That is, any twists or exceptional fibres should be negated before being added.
Regarding the signs of exceptional fibres: Consider a saturated block containing a solid torus whose meridinal curve runs p times horizontally around the boundary in order through annuli 0,1,... and follows the fibres q times from bottom to top (as depicted in the diagram in the NSatBlock class notes). Then this saturated block adds a positive (p, q) fibre to the underlying Seifert fibred space.
If the ring of saturated annuli bounding this block is twisted then the situation becomes more complex. It can be proven that such a block must contain a twisted reflector boundary in the base orbifold (use Z_2 homology with fibrereversing paths to show that the base orbifold must contain another twisted boundary component, and then recall that real boundaries are not allowed inside blocks).
In this twisted boundary case, it should be assumed that the twisted reflector boundary is already stored in the given Seifert fibred space. This routine should make any further changes that are required (there may well be none). That is, the space should be adjusted as though a trivial Seifert fibred space over the annulus with one twisted reflector boundary (and one twisted puncture corresponding to the block boundary) had been replaced by this block. In particular, this routine should not add the reflector boundary itself.
sfs  the Seifert fibred space to adjust. 
reflect  true if this block is to be reflected, or false if it should be inserted directly. 
Implemented in regina::NSatLayering, regina::NSatReflectorStrip, regina::NSatCube, regina::NSatTriPrism, regina::NSatLST, and regina::NSatMobius.

inline 
Returns details of the requested annulus on the boundary of this saturated block.
Annuli are numbered from 0 to nAnnuli()1 as described in the class notes.
which  indicates which boundary annulus is requested; this must be between 0 and nAnnuli()1 inclusive. 

pure virtual 
Returns a newly created clone of this saturated block structure.
A clone of the correct subclass of NSatBlock will be returned. For this reason, each subclass of NSatBlock must implement this routine.
Implemented in regina::NSatLayering, regina::NSatReflectorStrip, regina::NSatCube, regina::NSatTriPrism, regina::NSatLST, and regina::NSatMobius.
std::string regina::NSatBlock::getAbbr  (  bool  tex = false  )  const 
Returns an abbreviated name or symbol for this block.
This name will reflect the particular block type, but may not provide thorough details.
The name will be no more than a handful of characters long, and will not include a newline (or surrounding dollar signs in TeX mode).
tex  true if the name should be formatted for TeX, or false if it should be in plain text format. 

inline 
Returns whether there is another saturated block listed as being adjacent to the given boundary annulus of this block.
whichAnnulus  indicates which boundary annulus of this block should be examined; this must be between 0 and nAnnuli()1 inclusive. 
true
if the given boundary annulus has an adjacent block listed, or false
otherwise.

staticprotected 
Determines whether the given tetrahedron is contained within the given list.
This is intended as a helper routine for isBlock() and related routines.
t  the tetrahedron to search for. 
list  the list in which to search. 
true
if and only if the given tetrahedron was found.

inlinestaticprotected 
Determines whether the given tetrahedron is contained within the given list.
This is intended as a helper routine for isBlock() and related routines. It is a generic routine for working with arbitrary list types.
List::const_iterator
that span the given list can be obtained by calling list.begin()
and list.end()
.t  the tetrahedron to search for. 
list  the list in which to search. 
true
if and only if the given tetrahedron was found.

static 
Determines whether the given annulus is in fact a boundary annulus for a recognised type of saturated block.
The annulus should be represented from the inside of the proposed saturated block.
Only certain types of saturated block are recognised by this routine. More exotic saturated blocks will not be identified, and this routine will return null
in such cases.
The given list of tetrahedra will not be examined by this routine. That is, only saturated blocks that do not contain any of these tetrahedra will be considered. As a consequence, if the given annulus uses any of these tetrahedra then null
will be returned.
If a block is found on the other hand, all of the tetrahedra within this block will be added to the given list.
In the event that a block is found, it is guaranteed that the given annulus will be listed as annulus number 0 in the block structure, without any horizontal or vertical reflection.
annulus  the proposed boundary annulus that should form part of the new saturated block. 
avoidTets  the list of tetrahedra that should not be considered, and to which any new tetrahedra will be added. 
null
if none was found.

inline 
Returns the number of annuli on the boundary of this saturated block.
void regina::NSatBlock::nextBoundaryAnnulus  (  unsigned  thisAnnulus, 
NSatBlock *&  nextBlock,  
unsigned &  nextAnnulus,  
bool &  refVert,  
bool &  refHoriz,  
bool  followPrev  
) 
Finds the next (or previous) boundary annulus around from this, treating all adjacent blocks as part of a single large saturated region.
Suppose that all saturated blocks are merged together according to adjacent boundary annuli, forming larger saturated structures. The remaining annuli that do not have adjacent blocks will group together to form several large boundary rings. Note that each boundary ring might involve annuli from several different blocks, and might or might not have a twist (thus forming a large Klein bottle instead of a large torus).
This routine is used to trace around such a boundary ring. It is assumed that annulus thisAnnulus of this block forms part of a boundary ring (i.e., it has no adjacent block). This routine will then return the next/previous annulus around from this in the large boundary ring. Here "next" means in the direction following from the second triangle of this annulus, and "previous" means in the direction following from the first triangle; the boolean argument followPrev controls which we will be used. This next/previous annulus might belong to another block, or it might even be this original annulus again.
The next/previous annulus itself is not returned, but rather a reference as to how it appears within its enclosing saturated block. Specifically, a block and corresponding annulus number will be returned in the arguments nextBlock and nextAnnulus respectively.
It is possible that the next/previous annulus as it appears within the returned block is oriented differently from how it appears within this large boundary ring. For this reason, two booleans are returned also. The argument refVert will describe whether the annulus is reflected vertically as it appears within the large boundary ring (i.e., the first and second triangles remain the same but the fibre direction is reversed). Similarly, the argument refHoriz will describe whether the annulus is reflected horizontally as it appears within the large boundary ring (i.e., first and second triangles are switched but the fibre direction is unchanged).
It is possible that both a horizontal and vertical reflection take place. Note that any kind of reflection will also affect the locations of the 0/1/2 markings as described in the NSatAnnulus class notes.
Finally, note that if the large boundary ring is twisted (i.e., it forms a Klein bottle), then following the entire boundary ring around using this routine will bring you back to the starting annulus but with the refVert flag set.
thisAnnulus  describes which original boundary annulus of this block to examine; this must be between 0 and nAnnuli()1 inclusive. 
nextBlock  a reference used to return the block containing the next boundary annulus around from thisAnnulus. 
nextAnnulus  a reference used to return the specific annulus number within nextBlock of the next annulus around; this will be between 0 and nextBlock>nAnnuli()1 inclusive, and the corresponding annulus will have no block adjacent to it. 
refVert  a reference used to return true if the next annulus around is vertically reflected, or false if not; see above for details. 
refHoriz  a reference used to return true if the next annulus around is horizontally reflected, or false if not; see above for details. 
followPrev  true if we should find the previous boundary annulus, or false if we should find the next boundary annulus. 

inlinestaticprotected 
Determines whether the given tetrahedron pointer is null.
This is intended as a helper routine for isBlock() and related routines. Despite its trivial implementation, it is provided to make long blocks of code easier to read and distinguish by functionality.
The name notUnique() may seem strang for what is essentially a nullity test; in fact this routine is offered as a degenerate case of other variants of notUnique() that take more tetrahedra as arguments.
test  the tetrahedron pointer to test. 
true
if test is null, or false
otherwise.

inlinestaticprotected 
Determines whether the given tetrahedron pointer is null or equal to another from the given list.
This is intended as a helper routine for isBlock() and related routines. Despite its trivial implementation, it is provided to make long blocks of code easier to read and distinguish by functionality.
test  the tetrahedron pointer to test. 
other1  another tetrahedron that will be compared with test. 
true
if test is null or equal to other1, or false
otherwise.

inlinestaticprotected 
Determines whether the given tetrahedron pointer is null or equal to another from the given list.
This is intended as a helper routine for isBlock() and related routines. Despite its trivial implementation, it is provided to make long blocks of code easier to read and distinguish by functionality.
test  the tetrahedron pointer to test. 
other1  another tetrahedron that will be compared with test. 
other2  another tetrahedron that will be compared with test. 
true
if test is null or equal to other1 or other2, or false
otherwise.

inlinestaticprotected 
Determines whether the given tetrahedron pointer is null or equal to another from the given list.
This is intended as a helper routine for isBlock() and related routines. Despite its trivial implementation, it is provided to make long blocks of code easier to read and distinguish by functionality.
test  the tetrahedron pointer to test. 
other1  another tetrahedron that will be compared with test. 
other2  another tetrahedron that will be compared with test. 
other3  another tetrahedron that will be compared with test. 
true
if test is null or equal to other1, other2 or other3, or false
otherwise.

inlinestaticprotected 
Determines whether the given tetrahedron pointer is null or equal to another from the given list.
This is intended as a helper routine for isBlock() and related routines. Despite its trivial implementation, it is provided to make long blocks of code easier to read and distinguish by functionality.
test  the tetrahedron pointer to test. 
other1  another tetrahedron that will be compared with test. 
other2  another tetrahedron that will be compared with test. 
other3  another tetrahedron that will be compared with test. 
other4  another tetrahedron that will be compared with test. 
true
if test is null or equal to other1, other2, other3 or other4, or false
otherwise. bool regina::NSatBlock::operator<  (  const NSatBlock &  compare  )  const 
Implements a consistent ordering of saturated blocks.
This ordering is purely aesthetic on the part of the author, and is subject to change in future versions of Regina.
compare  the saturated block with which this will be compared. 
true
if this block comes before the given block according to the ordering of saturated blocks, or false
if either the blocks are identical or this block comes after the given block.

inline 
Lists the given saturated block as being adjacent to the given boundary annulus of this block.
Both block structures (this and the given block) will be updated.
whichAnnulus  indicates which boundary annulus of this block has the new adjacency; this must be between 0 and nAnnuli()1 inclusive. 
adjBlock  the other saturated block that is adjacent to this. 
adjAnnulus  indicates which boundary annulus of the adjacent block meets the given boundary annulus of this block; this must be between 0 and adjBlock>nAnnuli()1 inclusive. 
adjReflected  indicates whether the new adjacency is reflected (see the class notes for details). 
adjBackwards  indicates whether the new adjacency is backwards (see the class notes for details). 

virtual 
Adjusts the structure of this block according to the given isomorphism between triangulations.
Any triangulationspecific information will be transformed accordingly (for instance, the routine NSatAnnulus::transform() will be called for each boundary annulus).
Information regarding adjacent blocks will not be changed. Only structural information for this particular block will be updated.
The given isomorphism must describe a mapping from originalTri to newTri, and this block must currently refer to tetrahedra in originalTri. After this routine is called the block will instead refer to the corresponding tetrahedra in newTri (with changes in vertex/face numbering also accounted for).
originalTri  the triangulation currently used by this saturated block. 
iso  the mapping from originalTri to newTri. 
newTri  the triangulation to be used by the updated block structure. 
Reimplemented in regina::NSatLST.

inline 
Is the ring of boundary annuli twisted to form a long Mobius strip?
Recall from the class notes that the twist occurs between boundary annuli nAnnuli()1 and 0.
true
if the ring of boundary annuli is twisted, or false
if not.

pure virtual 
Writes an abbreviated name or symbol for this block to the given output stream.
This name should reflect the particular block type, but need not provide thorough details.
The output should be no more than a handful of characters long, and no newline should be written. In TeX mode, no leading or trailing dollar signs should be written.
out  the output stream to which to write. 
tex  true if the output should be formatted for TeX, or false if it should be in plain text format. 
Implemented in regina::NSatLayering, regina::NSatReflectorStrip, regina::NSatCube, regina::NSatTriPrism, regina::NSatLST, and regina::NSatMobius.

protected 
Describes which specific annulus of the adjacent saturated block is joined to each boundary annulus of this block.
Values may be undefined if the corresponding entries in the adjBlock array is null.

protected 
Describes whether the adjacency for each boundary annulus is backwards (see the class notes above).
Values may be undefined if the corresponding entries in the adjBlock array is null.

protected 
The saturated block joined to each boundary annulus; this may be null if there is no adjacency or if this information is not known.

protected 
Describes whether the adjacency for each boundary annulus is reflected (see the class notes above).
Values may be undefined if the corresponding entries in the adjBlock array is null.

protected 
Details of each boundary annulus, as seen from the inside of this saturated block.

protected 
The number of boundary annuli.

protected 
Is the ring of boundary annuli twisted to form a Mobius band?