Modification

There are many ways of modifying a 3-manifold triangulation. Many of these can be found in the various Triangulation menus, which appear when you open a triangulation for viewing.

Caution

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.

Editing Gluings (All Dimensions)

The simplest way to modify a triangulation is to open the Gluings tab and edit the facet gluings table directly. This allows you to change the gluings between triangles, tetrahedra or pentachora for 2-, 3- and 4-manifolds respectively. See the notes on viewing facet gluings for details on how to read the table.

You can add and remove triangles, tetrahedra or pentachora using the Add … and Remove … buttons above the table (the Add button is marked Add Triangle, Add Tet or Add Pent for 2-, 3- and 4-manifolds respectively, and likewise for the Remove button). To change the gluings between existing triangles, tetrahedra or pentachora, just type the new gluings directly into the table. If you want to remove a gluing (i.e., make a facet part of the triangulation boundary), simply delete the contents of the cell.

If you like, you can also name triangles, tetrahedra or pentachora to help keep track of their roles within the triangulation. Click on the cell in the leftmost column (containing the number that identifies that triangle, tetrahedron or pentachoron), and type a new name directly into the cell.

Automatic Simplification (3-D and 4-D)

Regina has a rich set of fast and effective moves for simplifying a triangulation without changing the underlying manifold. If you press the Simplify button (or select TriangulationSimplify), then Regina will use a combination of these moves to reduce the triangulation to as few tetrahedra (in 3-D) or pentachora (in 4-D) as it can [Bur13]. This procedure is always fast and often very effective, but there is no guarantee that this will produce the smallest possible triangulation: Regina might get stuck at a local minimum from which it cannot see how to escape.

If your triangulation has boundary, this routine will also try to make the number of boundary triangles (in 3-D) or boundary tetrahedra (in 4-D) as small as it can, though again there is no guarantee of reaching a global minimum.

Exhaustive Simplification (3-D)

For 3-manifolds, if Regina cannot simplify the triangulation immediately using its fast heuristics, then it will offer to perform a slower, more exhaustive search. Press the Try harder button, as shown below:

At this point, Regina will begin an exhaustive search through the Pachner graph (this is essentially the graph that relates all triangulations of the underlying manifold through 2-3 and 3-2 Pachner moves):

You can stop this search at any time by pressing the Cancel button.

Since the Pachner graph is infinite, Regina will only search up to a certain number of tetrahedra. If this still fails, it will offer to search further, and you can press Keep trying to continue the search using more tetrahedra than before. This cycle of ever-expanding searches will continue for as long as you like: Regina will only stop if you decline to keep searching, or you cancel a search while it is running, or it finds a smaller triangulation.

Warning

When searching the Pachner graph, both the time and memory consumption grow at a super-exponential rate relative to the number of tetrahedra. If you continue to press Keep trying over and over, you may well run out of time or memory (or both).

Manual Simplification: Elementary Moves (3-D and 4-D)

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 manifold. To perform elementary moves, select TriangulationElementary Move 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 a move, and then press Apply to perform it. You may continue to apply one move after another. When you are done, press Close to close the elementary move box.

We only give a brief summary of the various types of move here. The full details are in the API documentation, and for 3-manifolds you can also read about them in plain English [Bur13].

The available moves for 3-manifolds (as illustrated above) are:

3-2 Move

Replaces three tetrahedra joined along a degree 3 edge with two tetrahedra joined along a triangle.

2-3 Move

Replaces two tetrahedra joined along a triangle with three tetrahedra joined along a degree 3 edge.

1-4 Move

Replaces one tetrahedron with four tetrahedra that meet at a new internal degree 4 vertex.

4-4 Move

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.

2-0 Move (Edge)

Takes two tetrahedra joined along a degree 2 edge and squashes them flat.

2-0 Move (Vertex)

Takes two tetrahedra that meet at a degree 2 vertex and squashes them flat.

2-1 Move

Merges the tetrahedron containing a degree 1 edge with an adjacent tetrahedron.

Open Book

Takes an internal triangle with two boundary edges and “unglues” that triangle, creating two new boundary triangles and exposing the tetrahedra inside to the boundary.

Close Book

Folds together two adjacent boundary triangles around a common boundary edge, with the result of simplifying the boundary.

Shell Boundary

Removes an “unnecessary tetrahedron” that sits along the boundary of the triangulation.

Collapse Edge

Takes an edge between two distinct vertices and collapses it to a point. Any tetrahedra that contained the edge will be “flattened away”.

For 4-manifold triangulations, Regina offers a similar selection of moves, as illustrated below:

In brief, these 4-manifold moves are:

4-2 Move

Replaces four pentachora joined along a degree 4 edge with two pentachora glued together along a tetrahedron.

3-3 Move

Replaces three pentachora joined along a degree 3 triangle with three pentachora joined along a new degree 3 triangle that is aligned in a different direction.

2-4 Move

Replaces two pentachora glued together along a tetrahedron with four pentachora joined along a new degree 4 edge.

1-5 Move

Replaces one pentachoron with five pentachora that meet at a new internal degree 5 vertex.

2-0 Move (Triangle)

Takes two pentachora joined along a degree 2 triangle and squashes them flat.

2-0 Move (Edge)

Takes two pentachora that meet at a degree 2 edge and squashes them flat.

Open Book

Takes an internal tetrahedron with one, two or three boundary triangles and “unglues” that tetrahedron, creating two new boundary tetrahedra and exposing the pentachora inside to the boundary.

Shell Boundary

Removes an “unnecessary pentachoron” that sits along the boundary of the triangulation.

Collapse Edge

Takes an edge between two distinct vertices and collapses it to a point. Any pentachora that contained the edge will be “flattened away”.

0-Efficiency (3-D)

A 3-manifold 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 3-D TriangulationMake 0-Efficient.

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 orientable manifolds that cannot be made 0-efficient: RP3 and S2×S1. Regina will notify you if your triangulation represents one of these manifolds.

Caution

The algorithm to make a triangulation 0-efficient runs in worst-case exponential time, though it is often still extremely fast in practice. If your triangulation is large, you should consider whether automatic simplification will suffice: this is much faster at reducing the number of tetrahedra, and often produces a 0-efficient result. You can test the result for 0-efficiency via the recognition tab.

Switching Between Real and Ideal (3-D and 4-D)

You can convert between real boundary components (formed from boundary triangles in 3-D, or boundary tetrahedra in 4-D) and ideal boundary components (formed from individual vertices with closed non-spherical vertex links).

If you have an ideal triangulation, you can select TriangulationTruncate Ideal Vertices to convert your ideal vertices into real boundary components. Regina will subdivide the triangulation and delete a small neighbourhood of each ideal vertex. Any invalid vertices will be truncated also.

Tip

Because of the subdivision, this operation will greatly increase the size of the triangulation. 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 TriangulationMake Ideal.

Each real boundary component will be “coned” to a new ideal vertex: in 3-D this means adding one new tetrahedron for each boundary triangle, or in 4-D one new pentachoron for each boundary tetrahedron.

Your boundary components will all become ideal, but there are some caveats:

  • Your triangulation will contain new ideal vertices as described above, but also internal vertices whose links are spheres. To get rid of these internal vertices, try simplifying your triangulation.

  • Any spherical boundary components will disappear entirely; that is, they will be filled in with balls.

Subdivision (All Dimensions)

You can perform a barycentric subdivision on your triangulation by selecting TriangulationBarycentric Subdivision. In 2-D this involves splitting each original triangle into 6 = 3! smaller triangles, with new vertices added at the centroid of each original triangle and the centre of each original edge. In 3-D this involves splitting each original tetrahedron into 24 = 4! smaller tetrahedra, with new vertices at the centroid of each original tetrahedron, triangle and edge. In 4-D, each original pentachoron splits into 120 = 5! smaller pentachora, with new vertices added in a similar fashion.

Orienting Triangulations and Double Covers (All Dimensions)

If your triangulation is orientable but not oriented, you may wish to reorder the vertices of each top-dimensional simplex (e.g., vertices 0,1,2,3 of each tetrahedron in a 3-manifold) so that adjacent simplices have consistent orientations. To do this, press the Orient button (or select TriangulationOrient from the menu).

To convert a non-orientable triangulation into its orientable double cover, select TriangulationDouble Cover. If your triangulation has any orientable components, they will simply be duplicated.

Puncturing and Drilling (3-D)

You can puncture a 3-manifold triangulation; that is, remove a small ball from its interior and retriangulate. To do this, select 3-D TriangulationPuncture.

This will work correctly regardless of whether the triangulation is closed, ideal, and/or has real boundary triangles. There will be a new 2-sphere boundary, formed from two new boundary triangles.

You can also drill out a small regular neighbourhood of an edge of your triangulation. To do this, select 3-D TriangulationDrill Edge.

You will be asked which edge to drill out, as illustrated below. For each available edge, the drop-down list shows the edge number (as seen in the skeleton viewer), along with details of the individual tetrahedron edges that combine to form that edge of the triangulation.

Building Connected Sums (3-D)

You can combine two 3-manifold triangulations by forming their connected sum. This will convert some triangulation X into the connected sum X # Y for some other triangulation Y (note that Y is allowed to be the same as X). If both X and Y are oriented triangulations then the connected sum will respect these orientations, and will be oriented also.

The triangulation X must be one of Regina's native triangulation packets, since X will be modified directly. The triangulation Y may be either a native triangulation packet or a hybrid SnapPea triangulation, since Y will not be modified.

To form this connected sum, first open the the triangulation X for viewing, and then select 3-D TriangulationConnected Sum With from the menu.

Regina will ask you which other triangulation to sum with; in other words, Regina will ask you for the triangulation Y.

The triangulation X will be changed directly into the connected sum. The result will most likely contain multiple vertices, and you may wish to simplify the resulting triangulation before proceeding further.

Cutting Along and Crushing Normal Surfaces (3-D)

If you have a normal surface in a 3-manifold triangulation, you can either cut along your surface or crush it to a point.

  • Cutting along a surface involves carefully slicing along the surface and retriangulating the resulting polyhedra. The resulting triangulation will have new real boundary component(s) corresponding to the original surface.

    This operation has the advantages that it will never change the topology of the 3-manifold beyond the simple act of slicing along the surface, and it will never introduce ideal vertices or invalid edges.

    The main drawback is that it can vastly increase the total number of tetrahedra. This has severe implications if you need to do anything computationally intensive with the resulting triangulation.

  • Crushing a surface is a potentially destructive operation, but when used carefully can be extremely powerful. The crushing operation was originally described by Jaco and Rubinstein [JR03]; see [Bur14b] for a simplified treatment. In essence, the surface is first collapsed to a point, and then any non-tetrahedron pieces that remain are “flattened away”.

    One key advantage of crushing is that it always reduces the number of tetrahedra (unless you crush vertex links, in which case the triangulation stays the same).

    The main disadvantage is that it can change the topology of your triangulation, sometimes dramatically. For example, it can create ideal vertices, undo connected sums, change the genus of boundary components, and delete entire summands. In some cases it can even make your triangulation invalid (for instance, edges might become identified with themselves in reverse).

    You should only crush a surface when you have theoretical arguments that tell you exactly what might change and how to detect it. Examples of such arguments appear in [JR03], where crushing is used to great effect.

To cut along or crush a normal surface: open the list of normal surfaces, select your surface in the list, and then choose either Normal SurfacesCut Along Surface or Normal SurfacesCrush Surface.

Regina will create a new triangulation where the surface has been cut along or crushed accordingly. This new trianguation will appear beneath the normal surfaces in the packet tree. Your original triangulation will not be changed.

Note that you could end up with a disconnected triangulation. If so, you can extract connected components to work with one at a time.