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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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Vietoris-Rips complex homology of a higher degree than the ambient dimension
I have examples for $\ell_2$ distance in $\mathbb{R}^2$.
In particular, take $2n+2$ points at the vertices of a regular polygon of unit radius, and set the distance parameter to be slightly less than …