Finite topology as applied to image analysis
Computer Vision, Graphics, and Image Processing
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Our goal is to describe an economic way of presenting 3-manifolds numerically. The idea consists in replacing 3-manifolds by cell complexes (their special spines) and encoding the spines by strings of integers. The encoding is natural, i.e., it allows one to operate with manifolds without decoding. We describe an application of the encoding to computer enumeration of 3-manifolds and give the resulting table. A brief introduction into the theory of quantum invariants of 3-manifolds is also given. The invariants were used by the enumeration for automatic casting out of duplicates. Separately, we investigate 3-dimensional submanifolds of R3. Any such submanifold can be presented by a 3-dimensional binary picture. We give a criterion for a 3-dimensional binary picture to determine a 3-manifold.