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Computational topology is the study of decidability problems in topology and the algorithms that determine decidability. Examples of area of study include Normal Surface theory and the subproblems of unknot and $S^3$ recognition.
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Computational software in Algebraic Topology?
Sage allows you to play with simplicial complexes and their (co)homology.
http://www.sagemath.org/doc/reference/sage/homology/simplicial_complex.html
http://www.sagemath.org/doc/reference/sage/homol …