Regina Calculation Engine
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Represents a blocked Seifert fibred space with two boundary tori that are joined together. More...
#include <subcomplex/nblockedsfsloop.h>
Public Member Functions | |
~NBlockedSFSLoop () | |
Destroys this structure and its constituent components. More... | |
const NSatRegion & | region () const |
Returns details of the saturated region from which this triangulation is formed. More... | |
const NMatrix2 & | matchingReln () const |
Returns the matrix describing how the two torus boundaries of the saturated region are joined. More... | |
NManifold * | getManifold () 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 this object in long text format to the given output stream. More... | |
std::string | getName () const |
Returns the name of this specific triangulation as a human-readable string. More... | |
std::string | getTeXName () const |
Returns the name of this specific triangulation in TeX format. More... | |
virtual NAbelianGroup * | getHomologyH1 () 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 this object in short text format to the given output stream. More... | |
Input and Output | |
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 NBlockedSFSLoop * | isBlockedSFSLoop (NTriangulation *tri) |
Determines if the given triangulation is a blocked Seifert fibred space with identified boundaries, as described by this class. More... | |
static NStandardTriangulation * | isStandardTriangulation (NComponent *component) |
Determines whether the given component represents one of the standard triangulations understood by Regina. More... | |
static NStandardTriangulation * | isStandardTriangulation (NTriangulation *tri) |
Determines whether the given triangulation represents one of the standard triangulations understood by Regina. More... | |
Represents a blocked Seifert fibred space with two boundary tori that are joined together.
This is a particular type of triangulation of a graph manifold, formed from a single saturated region whose two torus boundaries are identified. An optional layering may be placed between the two torus boundaries to allow for a more interesting relationship between the two sets of boundary curves. For more detail on saturated regions and their constituent saturated blocks, see the NSatRegion class; for more detail on layerings, see the NLayering class.
The saturated region may have two boundary components formed from one saturated annulus each. Alternatively, it may have one boundary formed from two saturated annuli, where this boundary is pinched together so that each annulus becomes a two-sided torus (both of which are later joined together). None of the boundary components (or the two-sided tori discussed above) may be twisted (i.e., they must be tori, not Klein bottles).
The way in which the two torus boundaries are identified is specified by a 2-by-2 matrix, which expresses curves representing the fibres and base orbifold on the second boundary in terms of such curves on the first boundary (see the page on Notation for Seifert fibred spaces for terminology).
More specifically, suppose that f0 and o0 are directed curves on the first boundary torus and f1 and o1 are directed curves on the second boundary torus, where f0 and f1 represent the fibres of the region and o0 and o1 represent the base orbifold. Then the boundaries are joined according to the following relation:
[f1] [f0] [ ] = M * [ ] [o1] [o0]
If a layering is present between the two torus boundaries, then the corresponding boundary curves are not identified directly. In this case, the matrix M shows how the layering relates the curves on each 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 NStandardTriangulation routine getManifold() is implemented for this class, but getHomologyH1() is not.
regina::NBlockedSFSLoop::~NBlockedSFSLoop | ( | ) |
Destroys this structure and its constituent components.
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inherited |
Returns the output from writeTextLong() as a string.
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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 NTriangulation::getHomologyH1() 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 NStandardTriangulation. 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 NStandardTriangulation describes an entire NTriangulation (and not just a part thereof) then the results of this routine should be identical to the homology group obtained by calling NTriangulation::getHomologyH1() upon the associated real triangulation.
Reimplemented in regina::NLayeredSolidTorus, regina::NTriSolidTorus, regina::NLayeredChain, regina::NSpiralSolidTorus, regina::NLayeredTorusBundle, regina::NSnapPeaCensusTri, regina::NLayeredLensSpace, regina::NLayeredLoop, regina::NSnappedBall, regina::NTrivialTri, regina::NL31Pillow, and regina::NLayeredChainPair.
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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 NStandardTriangulation. 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::NStandardTriangulation.
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inherited |
Returns the name of this specific triangulation as a human-readable string.
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inherited |
Returns the name of this specific triangulation in TeX format.
No leading or trailing dollar signs will be included.
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static |
Determines if the given triangulation is a blocked Seifert fibred space with identified boundaries, as described by this class.
tri | the triangulation to examine. |
null
if the given triangulation is not of this form.
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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(NTriangulation*) may recognise more triangulations than this routine, since passing an entire triangulation allows access to more information.
component | the triangulation component under examination. |
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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(NComponent*), since passing an entire triangulation allows access to more information.
tri | the triangulation under examination. |
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inline |
Returns the matrix describing how the two torus boundaries of the saturated region are joined.
Note that if a layering is placed between the two boundary tori, 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.
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inline |
Returns details of the saturated region from which this triangulation is formed.
See the class notes above for further information.
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inherited |
Returns the output from writeTextShort() as a string.
__str__()
function.
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inlineinherited |
A deprecated alias for str(), which returns the output from writeTextShort() as a string.
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inlineinherited |
A deprecated alias for detail(), which returns the output from writeTextLong() as a string.
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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::NStandardTriangulation.
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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::NStandardTriangulation.
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virtual |
Writes this object in long text format to the given output stream.
The output should provide the user with all the information they could want. The output should be human-readable, should not contain extremely long lines (so users can read the output in a terminal), and should end with a final newline.
The default implementation of this routine merely calls writeTextShort() and adds a newline.
out | the output stream to which to write. |
Reimplemented from regina::ShareableObject.
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inlinevirtualinherited |
Writes this object in short text format to the given output stream.
The output should be human-readable, should fit on a single line, and should not end with a newline.
out | the output stream to which to write. |
Implements regina::ShareableObject.