3-Manifold Recognizer

In preceding chapters we described a number of important algorithms, which make heavy use of the Haken method of normal surfaces. As a rule, algorithms based on that method have exponential complexity and hence are impractical. In particular, although the recognition problem for Haken manifolds has an algorithmic solution, there is no chance of it being be realized by a computer program, at least in the foreseeable future. On the other hand, quite often experienced topologists recognize 3-manifolds rather quickly. They use other algorithms which, whether based on rigorous mathematics or intuition, are much more efficient. However, this gain does not come for free. The price is that one has to allow an algorithm to be only partial (i.e., not an algorithm at all in the formal meaning of this term). The problem of finding an efficient partial algorithm for answering a particular class of geometric questions is in itself a well-stated mathematical problem. Trying to solve it, we inevitably discover new structural properties of geometric objects.