![]() | Modification |
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There are many ways of modifying a 3-manifold triangulation. Many of these can be found in the menu, which appears when you open a triangulation for viewing.
If you open one triangulation for viewing but then select another in the packet tree, all modifications will apply to the triangulation that you have open for viewing.
The simplest way to modify a triangulation is to open the Gluings tab and edit the face gluings table directly. See the notes on viewing tetrahedron face gluings for details on how to read the table.
You can add and remove tetrahedra using the Add Tet and Remove Tet buttons, and you can change the gluings by typing directly into the table. If you want to remove a gluing (i.e., make a face part of the triangulation boundary), just delete the contents of the cell.
If you like, you can also name tetrahedra to help keep track of their roles within the triangulation. Click on the cell in the leftmost column (containing the tetrahedron number), and type a new name directly into the cell.
Regina has a rich set of fast and effective moves for simplifying a triangulation without changing the underlying 3-manifold. If you press the button (or select → ), then Regina will use a combination of these moves to reduce the triangulation to as few tetrahed ra as it can. This is often very effective, but there is no guarantee that this will produce the fewest possible tetrahedra: Regina might get stuck at a local minimum from which it cannot escape.
If your triangulation has boundary, this routine will also try to make the number of boundary faces as small as it can (but again there is no guarantee of reaching a global minimum).
Instead of using automatic simplification, you might wish to modify your triangulation manually one step at a time. You can do this using elementary moves, which are small local modifications to the triangulation that preserve the underlying 3-manifold. To perform an elementary move, select → from the menu.
This will bring up a box containing all the elementary moves that can be performed upon your triangulation. There are many different types of moves available, and this list may continue to grow with future releases of Regina.
For each type of move, you will be offered a drop-down list of locations at which the move can be performed. If a move is disabled (greyed out), this means there are no suitable locations in your triangulation for that move type. Select the type of move and its location, and press to perform the move.
We do not give full details of the various moves here; see
[Bur03] or the
NTriangulation
class notes in the calculation engine documentation
for full descriptions of the moves and restrictions on their
possible locations. A brief summary is as follows.
Replaces three tetrahedra joined along a degree 3 edge with two tetrahedra joined along a face.
Replaces two tetrahedra joined along a face with three tetrahedra joined along a degree 3 edge.
Replaces four tetrahedra joined along a degree 4 edge with four tetrahedra joined along a new degree 4 edge that points in a different direction.
Takes two tetrahedra joined along a degree 2 edge and squashes them flat.
Takes two tetrahedra that meet at a degree 2 vertex and squashes them flat.
Merges the tetrahedron containing a degree 1 edge with an adjacent tetrahedron.
Takes an internal face with two boundary edges and “unglues” that face, creating two new boundary faces and exposing the tetrahedra inside to the boundary.
Folds together two adjacent bou ndary faces around a common boundary edge, with the result of simplifying the boundary.
Removes an “unnecessary tetrahedron” that sits along the boundary of the triangulation.
Takes an edge between two distinct vertices and collapses it to a point. Any tetrahedra that contained the edge will be “flattened away”.
A triangulation is 0-efficient if its only normal spheres and discs are vertex linking, and if it has no 2-sphere boundary components [JR03]. 0-efficient triangulations have significant theoretical and practical advantages, and often use relatively few tetrahedra.
If your triangulation is closed, orientable and connected, you can convert it into a 0-efficient triangulation of the same 3-manifold by selecting → .
If your triangulation represents a composite 3-manifold then it cannot be made 0-efficient—in this case a full connected sum decomposition will be inserted beneath your triangulation in the packet tree, and your original triangulation will be left unchanged.
There are also two exceptional prime manifolds that cannot be made 0-efficient: RP3 and S2×S1. Regina will notify you if your triangulation represents one of these manifolds.
The algorithm to make a triangulation 0-efficient runs in worst-case exponential time. If your triangulation is large, you should consider whether automatic simplification will suffice (which is much faster at reducing the number of tetrahedra, but which does not guarantee a 0-efficient result).
You can convert between real boundary components (formed from boundary faces of tetrahedra) and ideal boundary components (formed from individual vertices with closed non-spherical vertex links).
If you have an ideal triangulation, you can select → to convert your ideal vertices into real boundary components. Regina will subdivide the triangulation and slide off a small neighbourhood of each ideal vertex. Any non-standard boundary vertices will be truncated also.
Because of the subdivision, this operation will greatly increase the number of tetrahedra. After you truncate ideal vertices, try simplifying your triangulation.
Conversely: if your triangulation has real boundary components and you wish to convert this into an ideal triangulation, select