| Decomposition |
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Recall the analysis of combinatorial composition described earlier. This analysis simply involves searching for well-structured features that Regina can recognise within a triangulation. For triangulations without any particularly nice structure, the combinatorial composition may give no information at all (many non-minimal triangulations are like this).
Here we describe some more complete decomposition tools that are available. These decomposition tools are guaranteed to work on any triangulation, regardless of whether it is minimal and/or particularly well-structured. The trade-off is that these decomposition tools often make heavy use of normal surfaces and can be quite slow for larger triangulations.
Regina can work with disconnected triangulations as well as connected triangulations. It may be desirable at times to break a large disconnected triangulation into smaller separate triangulations for its individual components. For this operation the -> menu item can be used.
Each (connected) component triangulation will be inserted as a new triangulation beneath the original in the packet tree. The original triangulation will remain unchanged.
Unlike most of the decomposition algorithms described in this section, component decomposition is very fast and should perform without problems for very large triangulations.
For closed orientable connected 3-manifold triangulations, Regina can in fact completely decompose the triangulation into a connected sum of prime 3-manifolds. The algorithm used is essentially the 0-efficiency algorithm of Jaco and Rubinstein [JR03], which in turn relies upon Rubinstein's 3-sphere recognition algorithm [Rub95] [Rub97].
This decomposition can be performed using the -> menu item. As before, a smaller prime triangulation for each summand will be inserted beneath the original triangulation in the packet tree, with the original triangulation remaining unchanged. It is guaranteed that each of the smaller prime triangulations will be 0-efficient (i.e., will have no non-vertex-linking normal spheres).
This procedure involves an analysis of normal surfaces and can be very slow for large triangulations.
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