Computational topology

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Algorithmic topology, or computational topology, is a subfield of topology with an overlap with areas of computer science, in particular computational complexity theory.

A primary concern of algorithmic topology, as its name suggests, is to develop efficient algorithms for solving topological problems. For example, an open problem, called the unknotting problem, is to find a polynomial time algorithm to see if a knot is unknotted.

Some parts of algorithmic topology can be considered experimental mathematics.

As the Workshop on Application of Topology in Science and Engineering September 18 - September 22, 2006, at the Mathematical Sciences Research Institute, Berkeley, CA has shown, applications of computational topology are developing in the following directions: image analysis (see, e.g., digital topology) and image compression, surface reconstruction from point clouds and random data, high-dimensional data analysis including dimensionality reduction, dynamical systems, proteomics and DNA structure analysis, robotics.