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I'm basically following Inflations of ideal triangulations, by Jaco and Rubenstein, for this. The procedure would involve a few decisions being made, so I wanted to open up a discussion on the best way to approach this procedure. Assuming a normal vertex-linking surface is given, the following steps are needed:
Choose a frame. Giving the user a choice here seems like a good idea, as I don't think there is a guaranteed unique frame of minimal complexity. Otherwise, it seems we must just brute-force all possible frames?
Assign orientation and transverse directions to edges along each branch in the frame. This is just labelling, and we can use Regina's labelling for orientation. Given such an edge as the equivalence set of edges of tetrahedra, I am convinced that choosing a transverse direction is equivalent to choosing an ordering of the edges of tetrahedra such that there is a face identification which maps one tetrahedron edge onto the next, and each face identification is used at most once (essentially, do the sensible thing while following the ring of tetrahedra around the edge).
Determine configuration polygons. This is mostly routine. The only potentially tricky part is that, for each face which intersects the frame in 2 or 3 edges, we need to work out the order in which to add the tetrahedra which achieves the minimal number of crossings. I think we just get to brute force this, but I think for any reasonable triangulation that shouldn't cause issues. Maybe have some sort of warning if nTet > some_predefined_limit?
Inflate at faces, then at edges. If the above steps are complete, then this should be pre-determined and we just follow the algorithm.
As for the actual implementation, triangulating a vertex link is already implemented. The frame, therefore, is just a set of edges of this triangulation. This does sort of skip the normal surface workings of Regina, but it's fairly straight-forward to track the vertex involved through the process of inflation, so the new normal surface created is just the vertex linking surface of the same vertex in the new triangulation. I just don't think the normal surface machinery of Regina is useful for inflations.
From the frame, we need to build the configuration polygons. This is a matter of following the ring of tetrahedra around an edge to build up a marked planer polygon, and then sub-dividing according to the given algorithm from the paper.
At this point, the edges of the frame give the face identifications which are to be removed, and this is where new tetrahedra are inserted (inflation at a face), and then we use the configuration polygon to identify the remaining faces (inflation at an edge).
The text was updated successfully, but these errors were encountered:
I'm basically following Inflations of ideal triangulations, by Jaco and Rubenstein, for this. The procedure would involve a few decisions being made, so I wanted to open up a discussion on the best way to approach this procedure. Assuming a normal vertex-linking surface is given, the following steps are needed:
As for the actual implementation, triangulating a vertex link is already implemented. The frame, therefore, is just a set of edges of this triangulation. This does sort of skip the normal surface workings of Regina, but it's fairly straight-forward to track the vertex involved through the process of inflation, so the new normal surface created is just the vertex linking surface of the same vertex in the new triangulation. I just don't think the normal surface machinery of Regina is useful for inflations.
From the frame, we need to build the configuration polygons. This is a matter of following the ring of tetrahedra around an edge to build up a marked planer polygon, and then sub-dividing according to the given algorithm from the paper.
At this point, the edges of the frame give the face identifications which are to be removed, and this is where new tetrahedra are inserted (inflation at a face), and then we use the configuration polygon to identify the remaining faces (inflation at an edge).
The text was updated successfully, but these errors were encountered: