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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Translation of Haken's paper “Theorie der Normalflächen”

Haken's paper "Theorie der Normalflächen" is one of the formative papers in the development of normal surface theory and provides an algorithm for detecting the unknot. While there are now a variety ...
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Connected Sum Decomposition of a Knot

Given a composite knot, is it possible to decompose it in prime knots by an algorithm that runs in polynomial time?
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computing homology of subvarieties of Euclidean spaces by persistent homology

Let $M$ be a submanifold of the Euclidean space $\mathbb{R}^n$. Let $G$ be a finite group acting on $M$ freely. I want to compute the homology (or even the cohomology ring) of $M/G$. Suppose the ...