| Modification |
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Once a triangulation has been created, a variety of options are available for modifying it. Many of these modifications are found in the menu – note that these menu items will only appear when a triangulation is currently being viewed, and will be performed upon this triangulation (and not whatever is selected in the visual packet tree).
When viewing a triangulation, the Gluings tab allows the user to add or remove tetrahedra and manually edit the gluings between tetrahedron faces.
Tetrahedra are numbered beginning at 0; thus if a triangulation has three tetrahedra they will be numbered 0, 1 and 2. Tetrahedra can be added and removed using the Add Tet and Remove Tet buttons above the gluing editor.
The gluing editor itself contains a grid of tetrahedron faces. Each row represents a tetrahedron in the triangulation and each column represents one of the four tetrahedron faces. Tetrahedron vertices are numbered 0, 1, 2 and 3, and each tetrahedron face is identified by the three vertices that it contains.
Each cell of this grid identifies to what the corresponding face is
glued. If a cell is empty,
corresponding face is a boundary face and is glued to nothing at
all. Otherwise it will contain a string like
5 (312). This means that the face for that cell is
glued to face 312 of tetrahedron 5. In particular, if the cell in
question belongs to face 013 of tetrahedron 4, this means that
vertices 0, 1 and 3 of tetrahedron 4 are glued to vertices 3, 1 and
2 of tetrahedron 5 respectively. Thus the cell contents describe
the precise gluing permutation.
To change how the face is glued, click in the cell. If you are in direct edit mode, you can simply type a new gluing over top of the old one in the format described above. If you are in pop-up dialog mode, a button will appear; when this button is clicked a dialog will appear that allows a new gluing to be selected. The mode can be changed between direct edit and pop-up dialog in the triangulation options.
Tetrahedra can be optionally named to help keep track of their roles within a triangulation. To change the name of a tetrahedron, click on the cell containing the tetrahedron number on the left hand side of the grid. The new name can be typed directly into the cell.
The -> menu item will present a list of elementary moves that can be performed upon the triangulation that is currently being viewed. Note that since there are restrictions on when particular moves may be performed, some elementary moves might not be available.
An elementary move will never change the topology of the
3-manifold. The individual moves and their restrictions are
described in full detail in the NTriangulation
class notes in the calculation engine documentation,
and most are described with diagrams in Burton's PhD thesis
[Bur03] which is available from the
Regina website. A brief summary is provided below.
A 3-2 move about an edge of degree 3 involves replacing the three tetrahedra joined at that edge with two tetrahedra joined by a face.
A 2-3 move about a non-boundary face involves replacing the two tetrahedra joined at that face with three tetrahedra joined by an edge.
A 4-4 move about an edge of degree 4 involves replacing the four tetrahedra joined at that edge with four tetrahedra joined along a different edge in a different position.
A 2-0 move about an edge of degree 2 involves taking the two tetrahedra joined at that edge and squashing them flat.
A 2-0 move about a vertex of degree 2 involves taking the two tetrahedra joined at that vertex and squashing them flat.
A 2-1 move about an edge of degree 1 involves merging the tetrahedron containing that edge with one of the tetrahedra joining it.
A book opening move on a face that touches the boundary involves ungluing that face to create two new boundary faces and thus exposing the tetrahedra inside to the boundary.
A book closing move on a boundary edge involves folding together the two boundary faces on either side. The aim of this move is to simplify the boundary of the triangulation.
A boundary shelling move on a boundary tetrahedron involves simply removing that tetrahedron.
Collapsing an edge involves taking an edge between two distinct vertices and collapsing that edge to a point. Any tetrahedra containing that edge will be flattened into faces.
A few routines are available for performing standard subdivisions on triangulations.
The -> menu item will perform a barycentric subdivision on the triangulation currently being viewed. A barycentric subdivision involves taking each tetrahedron, adding new vertices at the centroid of the tetrahedron, the centroid of each face and the midpoint of each edge and joining them with edges to split the tetrahedron into 24 smaller tetrahedra.
The -> menu item will convert the triangulation currently being viewed to a finite triangulation (one with no ideal vertices). This is done by subdividing each tetrahedron and then truncating at the ideal vertices. The resulting triangulation has each ideal or non-standard vertex converted to a real boundary component (one made from boundary faces of tetrahedra).
The operation was called in previous versions of Regina. It was renamed to avoid confusion since operations are now available in both ideal-to-finite and finite-to-ideal directions.
Currently a relatively straightforward routine is available for simplifying a triangulation as far as possible without manual intervention. Selecting the -> menu item will use elementary moves to reduce the triangulation to a local minimum number of tetrahedra. Note that this is not guaranteed to produce the minimum number of tetrahedra required.
As well as performing obvious reductions, this routine tries random 4-4 moves as well as book opening and book closing moves if it cannot reduce the triangulation directly. It does not yet try increasing the size of the triangulation (such as through 2-3 moves) to escape a local minimum well in which it might be stuck; this is on the TODO list.
For triangulations with boundary, this routine will also try to make the number of boundary faces as small as possible (though again there are not guarantees of achieving a global minimum).
In cases where this faster simplification routine is ineffective, the more powerful but slower 0-efficiency conversion (described below) can be used.
Sometimes one wishes to extend a triangulation beyond its boundary, such as thickening the boundary, stretching it or adding new topological components.
The -> menu item performs one such task. When this menu item is selected, all real boundary components of the triangulation are converted into ideal boundary components.
Specifically, each boundary component that is formed from two or more boundary faces is converted into a single ideal vertex. This is done by gluing a tetrahedron to each boundary face, and then gluing these tetrahedra together to mirror the ways in which the boundary faces are joined. The new vertices opposite the boundary faces then become identified as a single ideal vertex.
One side-effect of this operation is that any spherical boundary components become filled in with balls (since the new vertex will have a spherical link, i.e., it will become an ordinary internal vertex). If the triangulation has no boundary faces (i.e., no real boundary components) then this operation does nothing.
A triangulation is 0-efficient if its only normal spheres and discs are vertex linking, and if it has no 2-sphere boundary components. Jaco and Rubinstein [JR03] prove that most minimal triangulations of closed orientable irreducible 3-manifolds are 0-efficient, and in general it can be observed that 0-efficient triangulations tend to use relatively small numbers of tetrahedra.
To convert a triangulation into a 0-efficient triangulation of the same underlying 3-manifold, the -> menu item can be used. Note that, like the global simplification routine described above, there is no guarantee that the minimum possible number of tetrahedra will be achieved.
0-efficiency conversion is currently only available for closed orientable triangulations. Furthermore, if the triangulation represents a composite 3-manifold then it is impossible to construct a 0-efficient triangulation – in this case a full connected sum decomposition will be inserted beneath this triangulation in the packet tree.
This procedure involves an analysis of normal surfaces and can be very slow for large triangulations. For a faster but perhaps less effective simplification of triangulations, see the global simplification routine described above.
The -> menu item will convert a triangulation to a double cover. Each non-orientable component will be converted to an orientable double cover, and each orientable component will simply be duplicated.
A triangulation can be modified by finding a normal surface within it, and either cutting along that surface or crushing it to a point. Each operation has its advantages and disadvantages:
Cutting along a surface will never change the topology of the 3-manifold beyond the simple act of slicing along the surface. However, it has the potential to vastly increase the number of tetrahedra in the triangulation.
Crushing a surface to a point will never increase the number of
tetrahedra (and generally will reduce it). However, crushing can
introduce additional unexpected changes to the topology of the
3-manifold, and in some cases can introduce ideal vertices or
invalid edges. For details see
NNormalSurface::crush()
In general you should only crush a surface to a point when there are theoretical reasons to know that this is safe. Examples of safe scenarios can be found in Jaco and Rubinstein's paper on 0-efficiency [JR03].
A normal surface can be cut along or crushed as follows. Open a list of normal surfaces, select the surface to operate upon, and invoke either the -> or the -> menu item (note that these menu items only appear when viewing a normal surface list). A new triangulation will be created in which the surface has been cut along or crushed accordingly. The original triangulation will not be changed.
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