Regina Calculation Engine

A degenerate zerotetrahedron saturated block that corresponds to attaching a Mobius band to a single annulus boundary. More...
#include <subcomplex/nsatblocktypes.h>
Public Member Functions  
NSatMobius (const NSatMobius &cloneMe)  
Constructs a clone of the given block structure. More...  
int  position () const 
Describes how the Mobius band is attached to the boundary annulus. More...  
virtual NSatBlock *  clone () const 
Returns a newly created clone of this saturated block structure. More...  
virtual void  adjustSFS (NSFSpace &sfs, bool reflect) const 
Adjusts the given Seifert fibred space to insert the contents of this saturated block. More...  
virtual void  writeTextShort (std::ostream &out) const 
Writes this object in short text format to the given output stream. More...  
virtual void  writeAbbr (std::ostream &out, bool tex=false) const 
Writes an abbreviated name or symbol for this block to the given output stream. More...  
Public Member Functions inherited from regina::NSatBlock  
NSatBlock (const NSatBlock &cloneMe)  
Creates a new clone of the given block. More...  
~NSatBlock ()  
Destroys all internal arrays. 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  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...  
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  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 NSatMobius *  isBlockMobius (const NSatAnnulus &annulus, TetList &avoidTets) 
Determines whether the given annulus is a boundary annulus for a block of this type (Mobius band). More...  
Static Public Member Functions inherited from regina::NSatBlock  
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...  
Additional Inherited Members  
Public Types inherited from regina::NSatBlock  
typedef std::set< NTetrahedron * >  TetList 
The data structure used to store a list of tetrahedra that should not be examined by isBlock(). More...  
Protected Member Functions inherited from regina::NSatBlock  
NSatBlock (unsigned nAnnuli, bool twistedBoundary=false)  
Constructor for a block with the given number of annuli on the boundary. More...  
Static Protected Member Functions inherited from regina::NSatBlock  
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 inherited from regina::NSatBlock  
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...  
A degenerate zerotetrahedron saturated block that corresponds to attaching a Mobius band to a single annulus boundary.
This is a degenerate case of the layered solid torus (see the class NSatLST), where instead of joining a solid torus to an annulus boundary we join a Mobius band. The Mobius band can be thought of as a zerotetrahedron solid torus with two boundary triangles, which in fact are opposite sides of the same triangle. By attaching a zerotetrahedron Mobius band to an annulus boundary, we are effectively joining the two triangles of the annulus together.
The meridinal disc of this zerotetrahedron solid torus meets the three edges of the annulus in 1, 1 and 2 places, so it is in fact a degenerate (1,1,2) layered solid torus. Note that the weight 2 edge is the boundary edge of the Mobius strip.

inline 
Constructs a clone of the given block structure.
cloneMe  the block structure to clone. 

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. 
Implements regina::NSatBlock.

inlinevirtual 
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.
Implements regina::NSatBlock.

static 
Determines whether the given annulus is a boundary annulus for a block of this type (Mobius band).
This routine is a specific case of NSatBlock::isBlock(); see that routine for further details.
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 
Describes how the Mobius band is attached to the boundary annulus.
The class notes discuss the weight two edge of the Mobius band (or equivalently the boundary edge of the Mobius band). The return value of this routine indicates which edge of the boundary annulus this weight two edge is joined to.
In the NSatAnnulus class notes, the three edges of the annulus are denoted vertical, horizontal and boundary, and the vertices of each triangle are given markings 0, 1 and 2.
The return value of this routine takes the value 0, 1 or 2 as follows:

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. 
Implements regina::NSatBlock.

virtual 
Writes this object in short text format to the given output stream.
The output should be humanreadable, should fit on a single line, and should not end with a newline.
out  the output stream to which to write. 
Implements regina::ShareableObject.