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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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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What is $\pi_{23}(S^2)$?
The first $22$ homotopy groups of the $2$-sphere were worked out by Toda in 1962, but I cannot find any results extending that to any higher homotopy groups of $S^2$.
Are any more of these groups ...
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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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Upper bounds on the Gromov–Hausdorff distance using persistent homology
In persistent homology theory, stability theorems are important to show that the topological signatures extracted are stable under small changes. A key result is the following bound on the bottleneck ...
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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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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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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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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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Homeomorphic extension of a discrete function
Let $f : \{ 0,1 \} ^ {n} \rightarrow \{ 0,1 \} ^ {n}$ be a bijective map. Then is there a known computable way to extend it to a homeomorphism $g:[ 0,1 ] ^ {n} \rightarrow [ 0,1 ] ^ {n}?$
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Knot Diffie–Hellman
Here's an idea for a knot-based Diffie–Hellman exchange:
Public: random (oriented) knot $P$.
Private: random (oriented) knots $A$ and $B$.
Exchange: Alice sends (randomized or canonical ...
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Are there any non-elementary functions that are computable?
Does a function $\mathit{f}:\mathbb{R}→\mathbb{R}$ being non-elementary (not expressible as a combination of finitely many elementary operations), imply that it is not computable?
The particular case ...
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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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Is there an algorithm for the genus of a knot?
A Seifert surface of a knot is a surface whose boundary is the knot. The genus of a knot is the minimal genus among all the Seifert surfaces of the knot. My question is, is any algorithm known to ...
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Are hyperbolic $n$-manifolds recursively enumerable?
Fixing a dimension $n \ge 4$, is the class of closed hyperbolic $n$-manifolds recursively enumerable?
Since hyperbolic manifolds are triangulable I can reformulate this in the following more explicit ...
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