Regina Calculation Engine
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Describes a triangulation of a graph manifold formed by joining a bounded saturated region with a thin I-bundle over the torus, possibly with layerings in between. More...
#include <subcomplex/npluggedtorusbundle.h>
Public Member Functions | |
~NPluggedTorusBundle () | |
Destroys this structure and its constituent components. | |
const NTxICore & | bundle () const |
Returns an isomorphic copy of the thin I-bundle that forms part of this triangulation. | |
const NIsomorphism & | bundleIso () const |
Returns an isomorphism describing how the thin I-bundle forms a subcomplex of this triangulation. | |
const NSatRegion & | region () const |
Returns the saturated region that forms part of this triangulation. | |
const NMatrix2 & | matchingReln () const |
Returns the matrix describing how the two torus boundaries of the saturated region are joined by the thin I-bundle and layerings. | |
NManifold * | getManifold () const |
Returns the 3-manifold represented by this triangulation, if such a recognition routine has been implemented. | |
std::ostream & | writeName (std::ostream &out) const |
Writes the name of this triangulation as a human-readable string to the given output stream. | |
std::ostream & | writeTeXName (std::ostream &out) const |
Writes the name of this triangulation in TeX format to the given output stream. | |
void | writeTextLong (std::ostream &out) const |
Writes this object in long text format to the given output stream. | |
Static Public Member Functions | |
static NPluggedTorusBundle * | isPluggedTorusBundle (NTriangulation *tri) |
Determines if the given triangulation is a saturated region joined to a thin I-bundle via optional layerings, as described in the class notes above. |
Describes a triangulation of a graph manifold formed by joining a bounded saturated region with a thin I-bundle over the torus, possibly with layerings in between.
The thin I-bundle must be untwisted, so that it forms the product T x I
with two boundary tori. Moreover, it must be isomorphic to some existing instance of the class NTxICore.
The saturated region is described by an object of the class NSatRegion. This region must have precisely two boundary annuli. These may be two separate torus boundaries (each formed from its own saturated annulus). Alternatively, the saturated region may have a single boundary formed from both saturated annuli, where this boundary is pinched together so that each annulus becomes its own two-sided torus.
Either way, the saturated region effectively has two torus boundaries, each formed from two faces of the triangulation. These boundaries are then joined to the two torus boundaries of the thin I-bundle, possibly with layerings in between (for more detail on layerings, see the NLayering class). This is illustrated in the following diagram, where the small tunnels show where the torus boundaries are joined (possibly via layerings).
/--------------------\ /-----------------\ | ----- | | ----- | | Saturated region | | Thin I-bundle | | ----- | | ----- | \--------------------/ \-----------------/
The effect of the thin I-bundle and the two layerings is essentially to join the two boundaries of the saturated region according to some non-trivial homeomorphism of the torus. This homeomorphism is specified by a 2-by-2 matrix M as follows.
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 saturated region and o0 and o1 represent the base orbifold (see the page on Notation for Seifert fibred spaces for terminology). Then the torus boundaries of the saturated region are identified by the thin I-bundle and layerings according to the following relation:
[f1] [f0] [ ] = M * [ ] [o1] [o0]
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 more detail, writeTextLong() may be used instead.
The optional NStandardTriangulation routine getManifold() is implemented for this class, but getHomologyH1() is not.
Destroys this structure and its constituent components.
As an exception, the thin I-bundle is not destroyed, since it is assumed that this is referenced from elsewhere.
const NTxICore & regina::NPluggedTorusBundle::bundle | ( | ) | const [inline] |
Returns an isomorphic copy of the thin I-bundle that forms part of this triangulation.
Like all objects of class NTxICore, the thin I-bundle that is returned is an external object with its own separate triangulation of the product T x I
. For information on how the thin I-bundle is embedded within this triangulation, see the routine bundleIso().
const NIsomorphism & regina::NPluggedTorusBundle::bundleIso | ( | ) | const [inline] |
Returns an isomorphism describing how the thin I-bundle forms a subcomplex of this triangulation.
The thin I-bundle returned by bundle() does not directly refer to tetrahedra within this triangulation. Instead it contains its own isomorphic copy of the thin I-bundle triangulation (as is usual for objects of class NTxICore).
The isomorphism returned by this routine is a mapping from the triangulation bundle().core() to this triangulation, showing how the thin I-bundle appears as a subcomplex of this structure.
NManifold* regina::NPluggedTorusBundle::getManifold | ( | ) | const [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.
static NPluggedTorusBundle* regina::NPluggedTorusBundle::isPluggedTorusBundle | ( | NTriangulation * | tri | ) | [static] |
Determines if the given triangulation is a saturated region joined to a thin I-bundle via optional layerings, as described in the class notes above.
tri | the triangulation to examine. |
null
if the given triangulation is not of the form described by this class. const NMatrix2 & regina::NPluggedTorusBundle::matchingReln | ( | ) | const [inline] |
Returns the matrix describing how the two torus boundaries of the saturated region are joined by the thin I-bundle and layerings.
See the class notes above for details.
const NSatRegion & regina::NPluggedTorusBundle::region | ( | ) | const [inline] |
Returns the saturated region that forms part of this triangulation.
The region refers directly to tetrahedra within this triangulation (as opposed to the thin I-bundle, which refers to a separate external triangulation).
std::ostream& regina::NPluggedTorusBundle::writeName | ( | std::ostream & | out | ) | const [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.
std::ostream& regina::NPluggedTorusBundle::writeTeXName | ( | std::ostream & | out | ) | const [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.
void regina::NPluggedTorusBundle::writeTextLong | ( | std::ostream & | out | ) | const [virtual] |
Writes this object in long text format to the given output stream.
The output should provided the user with all the information they could want. The output should end with a 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.