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virtual | ~NTxICore () |
| Destroys this object. More...
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const NTriangulation & | core () const |
| Returns a full copy of the T x I triangulation that this object describes. More...
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unsigned | bdryTet (unsigned whichBdry, unsigned whichTri) const |
| Determines which tetrahedron provides the requested boundary triangle. More...
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NPerm4 | bdryRoles (unsigned whichBdry, unsigned whichTri) const |
| Describes which tetrahedron vertices play which roles in the upper and lower boundary triangles. More...
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const NMatrix2 & | bdryReln (unsigned whichBdry) const |
| Returns a 2-by-2 matrix describing the alpha and beta curves on a torus boundary in terms of specific tetrahedron edges. More...
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const NMatrix2 & | parallelReln () const |
| Returns a 2-by-2 matrix describing the parallel relationship between the upper and lower boundary curves. More...
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std::string | name () const |
| Returns the name of this specific triangulation of T x I as a human-readable string. More...
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REGINA_DEPRECATED std::string | getName () const |
| Deprecated routine that returns the name of this specific triangulation of T x I as a human-readable string. More...
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std::string | TeXName () const |
| Returns the name of this specific triangulation of T x I in TeX format. More...
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REGINA_DEPRECATED std::string | getTeXName () const |
| Deprecated routine that returns the name of this specific triangulation of T x I in TeX format. More...
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virtual std::ostream & | writeName (std::ostream &out) const =0 |
| Writes the name of this specific triangulation of T x I to the given output stream. More...
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virtual std::ostream & | writeTeXName (std::ostream &out) const =0 |
| Writes the name of this specific triangulation of T x I in TeX format to the given output stream. More...
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void | writeTextShort (std::ostream &out) const |
| Writes a short text representation of this object to the given output stream. More...
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void | writeTextLong (std::ostream &out) const |
| Writes a detailed text representation of this object to the given output stream. More...
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std::string | str () const |
| Returns a short text representation of this object. More...
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std::string | utf8 () const |
| Returns a short text representation of this object using unicode characters. More...
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std::string | detail () const |
| Returns a detailed text representation of this object. More...
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REGINA_DEPRECATED std::string | toString () const |
| A deprecated alias for str(). More...
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REGINA_DEPRECATED std::string | toStringLong () const |
| A deprecated alias for detail(). More...
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Provides a triangulation of the product T x I
(the product of the torus and the interval).
Generally these triangulations are only one tetrahedron thick (i.e., a "thin I-bundle"), though this is not a strict requirement of this class. Triangulations of this type are generally used as components of larger triangulations (such as layered surface bundles).
This product has two torus boundaries, called the upper and lower boundaries. Each of these boundary tori must be formed from precisely two triangles. This class tracks the mappings between parallel curves on the upper and lower boundaries, as well as mappings from boundary curves to specific tetrahedron edges.
For each of the two torus boundaries, two curves are chosen as generators of the fundamental group; these curves are called alpha and beta. Note that there is no requirement that the upper alpha and beta be parallel to the lower alpha and beta. The parallelReln() routine can be called to establish the precise relationship between these upper and lower curves.
Every object of this class contains a full copy of the triangulation that it describes (so you should not create excessive objects of this class without reason). This triangulation can be accessed through the core() routine.
const NMatrix2 & regina::NTxICore::bdryReln |
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unsigned |
whichBdry | ) |
const |
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inline |
Returns a 2-by-2 matrix describing the alpha and beta curves on a torus boundary in terms of specific tetrahedron edges.
Consider the first triangle of the given boundary. Let t be the tetrahedron returned by bdryTet(whichBdry, 0) and let p be the permutation returned by bdryRoles(whichBdry, 0).
Let edge01 be the directed edge from vertex p[0] to p[1] of tetrahedron t, and let edge02 be the directed edge from vertex p[0] to p[2] of tetrahedron t. Then the matrix returned by this routine describes how the directed edges edge01 and edge02 relate to the alpha and beta curves on the given boundary. Specifically:
[ alpha ] [ edge01 ]
[ ] = bdryReln() * [ ] .
[ beta ] [ edge02 ]
It is guaranteed that this matrix has determinant +1 or -1.
- Parameters
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whichBdry | 0 if the upper boundary should be examined, or 1 if the lower boundary should be examined. |
- Returns
- the relationship between the boundary curves and tetrahedron edges.
NPerm4 regina::NTxICore::bdryRoles |
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unsigned |
whichBdry, |
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unsigned |
whichTri |
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| const |
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inline |
Describes which tetrahedron vertices play which roles in the upper and lower boundary triangles.
Each boundary torus contains two triangles, whose vertices can be numbered 0, 1 and 2 according to the following diagram. This diagram is completely symmetric, in that edges 1-2 are no more special than edges 0-2 or 0-1. The important observations are that edges 1-2 and 2-1 of each triangle are identified, edges 0-2 and 2-0 of each triangle are identified and edges 0-1 and 1-0 of each triangle are identified.
*--->>--*
|0 2 / |
First | / 1| Second
triangle v / v triangle
|1 / |
| / 2 0|
*--->>--*
This routine returns a permutation that maps these integers 0,1,2 to real tetrahedron vertices. Let t be the tetrahedron returned by bdryTet(whichBdry, whichTri) and let p be the permutation returned by bdryRoles(whichBdry, whichTri). Then vertices p[0], p[1] and p[2] of tetrahedron t correspond to the markings 0, 1 and 2 respectively in the diagram above (and therefore the boundary triangle is face p[3] of the tetrahedron).
The arguments to this routine affect whether we examine the upper or lower boundary and whether we examine the first or second triangle of this boundary
- Parameters
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whichBdry | 0 if the upper boundary should be examined, or 1 if the lower boundary should be examined. |
whichTri | 0 if the first boundary triangle should be examined, or 1 if the second boundary triangle should be examined. |
- Returns
- the permutation mapping roles 0, 1 and 2 in the diagram above to real tetrahedron vertex numbers.
unsigned regina::NTxICore::bdryTet |
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unsigned |
whichBdry, |
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unsigned |
whichTri |
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) |
| const |
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inline |
Determines which tetrahedron provides the requested boundary triangle.
Recall that the T x I
triangulation has two torus boundaries, each consisting of two boundary triangles. This routine returns the specific tetrahedron that provides the given triangle of the given torus boundary.
What is returned is the index number of the tetrahedron within the triangulation. To access the tetrahedron itself, you may call core().tetrahedron(bdryTet(...))
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Note that the same tetrahedron may provide more than one boundary triangle.
- Parameters
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whichBdry | 0 if the upper boundary should be examined, or 1 if the lower boundary should be examined. |
whichTri | 0 if the first boundary triangle should be examined, or 1 if the second boundary triangle should be examined. |
const NMatrix2 & regina::NTxICore::parallelReln |
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const |
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inline |
Returns a 2-by-2 matrix describing the parallel relationship between the upper and lower boundary curves.
Let a_u and b_u be the upper alpha and beta boundary curves. Suppose that the lower alpha is parallel to w.a_u + x.b_u, and that the lower beta is parallel to y.a_u + z.b_u. Then the matrix returned will be
[ w x ]
[ ] .
[ y z ]
In other words, if a_l and b_l are the lower alpha and beta curves respectively, we have
[ a_l ] [ a_u ]
[ ] = parallelReln() * [ ] .
[ b_l ] [ b_u ]
- Returns
- the relationship between the upper and lower boundary curves.