Regina Calculation Engine
|
Represents a spiralled solid torus in a triangulation. More...
#include <subcomplex/nspiralsolidtorus.h>
Public Member Functions | |
virtual | ~NSpiralSolidTorus () |
Destroys this spiralled solid torus. More... | |
NSpiralSolidTorus * | clone () const |
Returns a newly created clone of this structure. More... | |
unsigned long | getNumberOfTetrahedra () const |
Returns the number of tetrahedra in this spiralled solid torus. More... | |
NTetrahedron * | getTetrahedron (unsigned long index) const |
Returns the requested tetrahedron in this spiralled solid torus. More... | |
NPerm4 | getVertexRoles (unsigned long index) const |
Returns a permutation represeting the role that each vertex of the requested tetrahedron plays in the solid torus. More... | |
void | reverse () |
Reverses this spiralled solid torus. More... | |
void | cycle (unsigned long k) |
Cycles this spiralled solid torus by the given number of tetrahedra. More... | |
bool | makeCanonical (const NTriangulation *tri) |
Converts this spiralled solid torus into its canonical representation. More... | |
bool | isCanonical (const NTriangulation *tri) const |
Determines whether this spiralled solid torus is in canonical form. More... | |
NManifold * | getManifold () const |
Returns the 3-manifold represented by this triangulation, if such a recognition routine has been implemented. More... | |
NAbelianGroup * | getHomologyH1 () const |
Returns the expected first homology group of this triangulation, if such a 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 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 NSpiralSolidTorus * | formsSpiralSolidTorus (NTetrahedron *tet, NPerm4 useVertexRoles) |
Determines if the given tetrahedron forms part of a spiralled solid torus with its vertices playing the given roles in the solid torus. 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 spiralled solid torus in a triangulation.
A spiralled solid torus is created by placing tetrahedra one upon another in a spiralling fashion to form a giant loop.
For each tetrahedron, label the vertices A, B, C and D. Draw the tetrahedron so that the vertices form an upward spiral in the order A-B-C-D, with D directly above A. Face BCD is on the top, face ABC is on the bottom and faces ABD and ACD are both vertical.
When joining two tetrahedra, face BCD of the lower tetrahedron will be joined to face ABC of the upper tetrahedron. In this way the tetrahedra are placed one upon another to form a giant loop (which is closed up by placing the bottommost tetrahedron above the topmost tetrahedron in a similar fashion), forming a solid torus overall.
In each tetrahedron, directed edges AB, BC and CD are major edges, directed edges AC and BD are minor edges and directed edge AD is an axis edge.
The major edges all combined form a single longitude of the solid torus. Using this directed longitude, using the directed meridinal curve ACBA and assuming the spiralled solid torus contains n tetrahedra, the minor edges all combined form a (2, n) curve and the axis edges all combined form a (3, n) curve on the torus boundary.
Note that all tetrahedra in the spiralled solid torus must be distinct and there must be at least one tetrahedron.
Note also that class NTriSolidTorus represents a spiralled solid torus with precisely three tetrahedra. A spiralled solid torus with only one tetrahedron is in fact a (1,2,3) layered solid torus.
All optional NStandardTriangulation routines are implemented for this class.
|
inlinevirtual |
Destroys this spiralled solid torus.
NSpiralSolidTorus* regina::NSpiralSolidTorus::clone | ( | ) | const |
Returns a newly created clone of this structure.
void regina::NSpiralSolidTorus::cycle | ( | unsigned long | k | ) |
Cycles this spiralled solid torus by the given number of tetrahedra.
Tetrahedra k, k+1, k+2 and so on will become tetrahedra 0, 1, 2 and so on respectively. Note that this operation will not change the vertex roles.
The underlying triangulation is not changed; all that changes is how this spiralled solid torus is represented.
k | the number of tetrahedra through which we should cycle. |
|
inherited |
Returns the output from writeTextLong() as a string.
|
static |
Determines if the given tetrahedron forms part of a spiralled solid torus with its vertices playing the given roles in the solid torus.
Note that the boundary triangles of the spiralled solid torus need not be boundary triangles within the overall triangulation, i.e., they may be identified with each other or with triangles of other tetrahedra.
tet | the tetrahedron to examine. |
useVertexRoles | a permutation describing the role each tetrahedron vertex must play in the solid torus; this must be in the same format as the permutation returned by getVertexRoles(). |
null
if the given tetrahedron is not part of a spiralled solid torus with the given vertex roles.
|
virtual |
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 from regina::NStandardTriangulation.
|
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.
|
inherited |
Returns the name of this specific triangulation as a human-readable string.
|
inline |
Returns the number of tetrahedra in this spiralled solid torus.
|
inline |
Returns the requested tetrahedron in this spiralled solid torus.
Tetrahedra are numbered from 0 to getNumberOfTetrahedra()-1 inclusive, with tetrahedron i+1 being placed above tetrahedron i.
index | specifies which tetrahedron to return; this must be between 0 and getNumberOfTetrahedra()-1 inclusive. |
|
inherited |
Returns the name of this specific triangulation in TeX format.
No leading or trailing dollar signs will be included.
|
inline |
Returns a permutation represeting the role that each vertex of the requested tetrahedron plays in the solid torus.
The permutation returned (call this p
) maps 0, 1, 2 and 3 to the four vertices of tetrahedron index so that vertices p[0]
, p[1]
, p[2]
and p[3]
correspond to vertices A, B, C and D respectively as described in the general class notes.
In particular, the directed edge from vertex p[0]
to p[3]
is an axis edge, directed edges p[0]
to p[2]
and p[1]
to p[3]
are minor edges and the directed path from vertices p[0]
to p[1]
to p[2]
to p[3]
follows the three major edges.
See the general class notes for further details.
index | specifies which tetrahedron in the solid torus to examine; this must be between 0 and getNumberOfTetrahedra()-1 inclusive. |
bool regina::NSpiralSolidTorus::isCanonical | ( | const NTriangulation * | tri | ) | const |
Determines whether this spiralled solid torus is in canonical form.
Canonical form is described in detail in the description for makeCanonical().
tri | the triangulation in which this solid torus lives. |
true
if and only if this spiralled solid torus is in canonical 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(NTriangulation*) 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(NComponent*), since passing an entire triangulation allows access to more information.
tri | the triangulation under examination. |
bool regina::NSpiralSolidTorus::makeCanonical | ( | const NTriangulation * | tri | ) |
Converts this spiralled solid torus into its canonical representation.
The canonical representation of a spiralled solid torus is unique in a given triangulation.
Tetrahedron 0 in the spiralled solid torus will be the tetrahedron with the lowest index in the triangulation, and under permutation getVertexRoles(0)
the image of 0 will be less than the image of 3.
tri | the triangulation in which this solid torus lives. |
true
if and only if the representation of this spiralled solid torus was actually changed. void regina::NSpiralSolidTorus::reverse | ( | ) |
Reverses this spiralled solid torus.
Tetrahedra 0, 1, 2, ..., getNumberOfTetrahedra()-1 will become tetrahedra getNumberOfTetrahedra()-1, ..., 2, 1, 0 respectively. Note that this operation will change the vertex roles as well.
The underlying triangulation is not changed; all that changes is how this spiralled solid torus is represented.
|
inherited |
Returns the output from writeTextShort() as a string.
__str__()
function.
|
inlineinherited |
A deprecated alias for str(), which returns the output from writeTextShort() as a string.
|
inlineinherited |
A deprecated alias for detail(), which returns the output from writeTextLong() as a string.
|
inlinevirtual |
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.
|
inlinevirtual |
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.
|
inlinevirtual |
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.
|
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.