Questions tagged [computational-topology]
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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Known obstruction for efficient computation of Stable homotopy groups?
Computation of stable homotopy groups (for example of sphere) is hard, but still, not as hard as unstable ones.
For unstable homotopy groups there are some results showing that there cannot be ...
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Computing the equivariant cohomology class of a Białynicki-Birula cell
One of my current research interests is Hessenberg varieties. Briefly, if $m_1\le m_2\le \cdots \le m_{n-1}$ is a weakly increasing sequence of positive integers such that $i\le m_i\le n$ for all $i$, ...
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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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Coarsifying persistence modules
The context
Let $I=[0,∞)$ and consider the category of persistence modules $(V,π)$ indexed over $I$ satisfying:
For all $t$ in $I$ but a closed discrete set of points $T$, there exists a ...
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Hochschild cohomology of path algebra as a cohomology of simplicial complex
M. Gerstenhaber and S. D. Schack have shown that a cohomology of simplicial complex can be expressed as a Hochschild cohomology of path algebra constructed from this complex (link).
Is the opposite ...
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KLO for operations over braids
KLO is a program that permits you to the do twistings, band operations over knots or Kirby diagram. However, I couldn't find a function on KLO that permits me to do the same thing over braids.
Is ...
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What field of mathematics is this? Necessary and sufficient corridors for topological routing
I am a computer scientist working on a problem in electronics design. The overall problem is about how to route traces on a circuit board, and I am looking for help on one of the subproblems.
We ...
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Is finding boundary-reducing discs for PL 3-manifolds with boundary pattern computationally efficient?
I am working with manifolds with boundary pattern, as defined by Matveev in his book Algorithmic Topology and Classification of 3-Manifolds. A manifold with boundary pattern is a pair $(M, P)$ where $...
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Motivation for persistent homology with respect to eigenfunctions of distance kernel operator in a recent preprint
I have a question about a recent preprint https://arxiv.org/pdf/1912.02225.pdf by Maria, Oudot, and Solomon. As far as I understand, in Section 8 they prove that persistent homology (persistence ...
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The complexity of cutting hackers in a computer network
Let $l_1,l_2,\dots,l_m$ be parallel lines in the plane, say $l_k=\mathbb R\times\{k\}$. On the $k$th line fix a set $V_k$ consisting of $n_k$ points.
Let $(V,E)$ be a directed graph whose set of ...
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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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Description of a point cloud being "undersampled" wrt persistent homology, confidence level?
I am completely new to topological data analysis, so I apologise if this is a well-known area of persistent homology, as well as for any imprecise language.
Suppose we know completely the topological ...
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If a set is covered by simplices then can it be covered by "almost disjoint" simplices?
Let $x_1,\dots,x_N$ be points in Euclidean space $\mathbb{R}^d$ (positive $d$), $r>0$, and consider set $X\subset\mathbb{R}^d$ defined as the collection of all $x\in \mathbb{R}^d$ of the form
$$
x =...
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Discrete Morse theory, choice of Morse function, and removing noise
If I have a simplicial complex, and a discrete Morse function defined on the simplices, I can use persistent homology to produce a barcode which helps me distinguish "persistent" shape from noise. To ...
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Local to global complexity of triangulations
Alright 3rd time's the charm - editing again to put all my cards on the table.
Consider a PL $n$-manifold $M$. Define the complexity $c(M)$ of $M$ to be the minimum number of $n$-simplices needed to ...
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(Best) ways to reduce knot complexity?
Lets say I have a diagram of a knot in some notation.
What's the fastest algorithm to simplify it? Or asked differently: what algorithms do software usually use?
I do not need to put it into the very ...
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Direct representation of simplical complexes in a HoTT implementation
Persistent homology can be used to transform a point-cloud into a simplical complex.
Do such simplical complexes have a first-class representation:
Conceptually, within HoTT?
Concretely, within some ...
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Two-parameter “$\varepsilon$-$\delta$ filtration” given a function between metric spaces
Let $X,Y$ be metric space and $f : X \to Y$ a (not necessarily continuous) function. I'm interested in the two-parameter filtration $(X_{\varepsilon, \delta})_{{\varepsilon, \delta} > 0}$ where $X_{...
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Computational tasks resulting from Chern-Weil theory
I have recently learned Chern-Weil theory for smooth and complex manifolds, as well as surrounding material on cohomology with integral coefficients.
I am curious what computational tasks are ...
user30211
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Persistent diagrams for images : existing implementations or packages?
I am interest to compute the persistent diagram associated to the image of a persistent module as in ''Persistent Homology for Kernels, Images, and Cokernels'' : https://epubs.siam.org/doi/epdf/10....
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