Regina Calculation Engine
|
Represents a blocked pair of Seifert fibred spaces joined along a single connecting torus. More...
#include <subcomplex/blockedsfspair.h>
Public Member Functions | |
~BlockedSFSPair () | |
Destroys this structure and its constituent components. More... | |
const SatRegion & | region (int which) const |
Returns details of one of the two bounded saturated regions that form this triangulation. More... | |
const Matrix2 & | matchingReln () const |
Returns the matrix describing how the two saturated region boundaries are joined. More... | |
Manifold * | manifold () const |
Returns the 3-manifold represented by this triangulation, if such a recognition routine has been implemented. More... | |
std::ostream & | writeName (std::ostream &out) const |
Writes the name of this triangulation as a human-readable string to the given output stream. More... | |
std::ostream & | writeTeXName (std::ostream &out) const |
Writes the name of this triangulation in TeX format to the given output stream. More... | |
void | writeTextLong (std::ostream &out) const |
Writes a detailed text representation of this object to the given output stream. More... | |
std::string | name () const |
Returns the name of this specific triangulation as a human-readable string. More... | |
std::string | TeXName () const |
Returns the name of this specific triangulation in TeX format. More... | |
virtual AbelianGroup * | homology () const |
Returns the expected first homology group of this triangulation, if such a routine has been implemented. More... | |
AbelianGroup * | homologyH1 () const |
Returns the expected first homology group of this triangulation, if such a routine has been implemented. More... | |
virtual void | writeTextShort (std::ostream &out) const |
Writes a short text representation of this object to the given output stream. More... | |
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... | |
Static Public Member Functions | |
static BlockedSFSPair * | isBlockedSFSPair (Triangulation< 3 > *tri) |
Determines if the given triangulation is a blocked pair of Seifert fibred spaces, as described by this class. More... | |
static StandardTriangulation * | isStandardTriangulation (Component< 3 > *component) |
Determines whether the given component represents one of the standard triangulations understood by Regina. More... | |
static StandardTriangulation * | isStandardTriangulation (Triangulation< 3 > *tri) |
Determines whether the given triangulation represents one of the standard triangulations understood by Regina. More... | |
Represents a blocked pair of Seifert fibred spaces joined along a single connecting torus.
This is a particular type of triangulation of a graph manifold, formed from two saturated regions whose torus boundaries are identified. An optional layering may be placed between the two torus boundaries to allow for a more interesting relationship between the boundary curves of each region. For more detail on saturated regions and their constituent saturated blocks, see the SatRegion class; for more detail on layerings, see the Layering class.
Each of the two saturated regions must have precisely one boundary component formed from just one saturated annulus, and this boundary may not be twisted (i.e., it must be a torus, not a Klein bottle). The way in which the boundaries from each region are identified is specified by a 2-by-2 matrix M, which expresses curves representing the fibres and base orbifold of the second region in terms of the first (see the page on Notation for Seifert fibred spaces for terminology).
More specifically, suppose that f0 and o0 are directed curves on the first region boundary and f1 and o1 are directed curves on the second region boundary, where f0 and f1 represent the fibres of each region and o0 and o1 represent the base orbifolds. Then the boundaries are joined according to the following relation:
[f1] [f0] [ ] = M * [ ] [o1] [o0]
If a layering is present between the two boundaries, then the boundary curves are not identified directly. In this case, the matrix M shows how the layering relates the curves on each region boundary.
Note that the routines writeName() and writeTeXName() do not offer enough information to uniquely identify the triangulation, since this essentially requires 2-dimensional assemblings of saturated blocks. For full details, writeTextLong() may be used instead.
The optional StandardTriangulation routine manifold() is implemented for this class, but homology() is not.
regina::BlockedSFSPair::~BlockedSFSPair | ( | ) |
Destroys this structure and its constituent components.
|
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.
|
inlinevirtualinherited |
Returns the expected first homology group of this triangulation, if such a routine has been implemented.
If the calculation of homology has not yet been implemented for this triangulation then this routine will return 0.
This routine does not work by calling Triangulation<3>::homology() on the associated real triangulation. Instead the homology is calculated directly from the known properties of this standard triangulation.
The details of which standard triangulations have homology calculation routines can be found in the notes for the corresponding subclasses of StandardTriangulation. The default implementation of this routine returns 0.
The homology group will be newly allocated and must be destroyed by the caller of this routine.
If this StandardTriangulation describes an entire Triangulation<3> (and not just a part thereof) then the results of this routine should be identical to the homology group obtained by calling Triangulation<3>::homology() upon the associated real triangulation.
This routine can also be accessed via the alias homologyH1() (a name that is more specific, but a little longer to type).
Reimplemented in regina::LayeredSolidTorus, regina::TriSolidTorus, regina::LayeredChain, regina::SpiralSolidTorus, regina::LayeredTorusBundle, regina::SnapPeaCensusTri, regina::LayeredLensSpace, regina::SnappedBall, regina::LayeredLoop, regina::TrivialTri, regina::L31Pillow, and regina::LayeredChainPair.
|
inlineinherited |
Returns the expected first homology group of this triangulation, if such a routine has been implemented.
If the calculation of homology has not yet been implemented for this triangulation then this routine will return 0.
This routine does not work by calling Triangulation<3>::homology() on the associated real triangulation. Instead the homology is calculated directly from the known properties of this standard triangulation.
The details of which standard triangulations have homology calculation routines can be found in the notes for the corresponding subclasses of StandardTriangulation. The default implementation of this routine returns 0.
The homology group will be newly allocated and must be destroyed by the caller of this routine.
If this StandardTriangulation describes an entire Triangulation<3> (and not just a part thereof) then the results of this routine should be identical to the homology group obtained by calling Triangulation<3>::homology() upon the associated real triangulation.
This routine can also be accessed via the alias homology() (a name that is less specific, but a little easier to type).
|
static |
Determines if the given triangulation is a blocked pair of Seifert fibred spaces, as described by this class.
tri | the triangulation to examine. |
null
if the given triangulation is not of this form.
|
staticinherited |
Determines whether the given component represents one of the standard triangulations understood by Regina.
The list of recognised triangulations is expected to grow between releases.
If the standard triangulation returned has boundary triangles then the given component must have the same corresponding boundary triangles, i.e., the component cannot have any further identifications of these boundary triangles with each other.
Note that the triangulation-based routine isStandardTriangulation(Triangulation<3>*) may recognise more triangulations than this routine, since passing an entire triangulation allows access to more information.
component | the triangulation component under examination. |
|
staticinherited |
Determines whether the given triangulation represents one of the standard triangulations understood by Regina.
The list of recognised triangulations is expected to grow between releases.
If the standard triangulation returned has boundary triangles then the given triangulation must have the same corresponding boundary triangles, i.e., the triangulation cannot have any further identifications of these boundary triangles with each other.
This routine may recognise more triangulations than the component-based isStandardTriangulation(Component<3>*), since passing an entire triangulation allows access to more information.
tri | the triangulation under examination. |
|
virtual |
Returns the 3-manifold represented by this triangulation, if such a recognition routine has been implemented.
If the 3-manifold cannot be recognised then this routine will return 0.
The details of which standard triangulations have 3-manifold recognition routines can be found in the notes for the corresponding subclasses of StandardTriangulation. The default implementation of this routine returns 0.
It is expected that the number of triangulations whose underlying 3-manifolds can be recognised will grow between releases.
The 3-manifold will be newly allocated and must be destroyed by the caller of this routine.
Reimplemented from regina::StandardTriangulation.
|
inline |
Returns the matrix describing how the two saturated region boundaries are joined.
Note that if a layering is placed between the two region boundaries, then any changes to the boundary relationships caused by the layering are included in this matrix.
See the class notes above for precise information on how this matrix is presented.
|
inherited |
Returns the name of this specific triangulation as a human-readable string.
|
inline |
Returns details of one of the two bounded saturated regions that form this triangulation.
See the class notes above for further information.
which | 0 if the first region should be returned, or 1 if the second region should be returned. |
|
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.
__str__()
.
|
inherited |
Returns the name of this specific triangulation in TeX format.
No leading or trailing dollar signs will be included.
|
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.
|
virtual |
Writes the name of this triangulation as a human-readable string to the given output stream.
out | the output stream to which to write. |
Implements regina::StandardTriangulation.
|
virtual |
Writes the name of this triangulation in TeX format to the given output stream.
No leading or trailing dollar signs will be included.
out | the output stream to which to write. |
Implements regina::StandardTriangulation.
|
virtual |
Writes a detailed text representation of this object to the given output stream.
This may be reimplemented by subclasses, but the parent StandardTriangulation class offers a reasonable default implementation based on writeName().
out | the output stream to which to write. |
Reimplemented from regina::StandardTriangulation.
|
inlinevirtualinherited |
Writes a short text representation of this object to the given output stream.
This may be reimplemented by subclasses, but the parent StandardTriangulation class offers a reasonable default implementation based on writeName().
out | the output stream to which to write. |