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.

Filter by
Sorted by
Tagged with
0 votes
0 answers
43 views

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_{...
0 votes
0 answers
58 views

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 ...
14 votes
1 answer
841 views

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 ...
1 vote
0 answers
185 views

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 ...
4 votes
1 answer
341 views

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 ...
3 votes
0 answers
125 views

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 ...
2 votes
0 answers
91 views

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 ...
4 votes
0 answers
93 views

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 ...
0 votes
0 answers
42 views

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....
3 votes
1 answer
158 views

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 $g:[ 0,1 ] ^ {n} \rightarrow [ 0,1 ] ^ {n}?$
8 votes
2 answers
410 views

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 ...
-2 votes
1 answer
230 views

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 ...
32 votes
4 answers
7k views

Computational software in Algebraic Topology?

I was wondering if there is any good software out there that allows you to do specific computations in algebraic topology. For example: Create a simplicial complex/set and ask questions about its ...
14 votes
4 answers
2k views

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 ...
2 votes
0 answers
54 views

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 =...
10 votes
6 answers
1k views

Homology software

What software is there to efficiently compute homology? Specifically: What software can take a simplicial complex (provided by a file listing maximal simplices, for example) and quickly compute its ...
15 votes
1 answer
351 views

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 ...
10 votes
3 answers
1k views

Deep learning for knot theory. Classification

As far as I know, there is a classification of all prime knots with less than 16 crossings. It seems that there is already a fast enough algorithm to distinguish a knot from an unknot. So in principle ...
11 votes
1 answer
249 views

Algorithmic Borel finiteness for hyperbolic manifolds

It is a theorem of Borel that there is a finite number of arithmetic hyperbolic manifolds of volume bounded above by $V.$ Is there any algorithm (or hope of an algorithm) to actually construct all of ...
6 votes
1 answer
534 views

Properties a triangulation must have in order to describe a manifold

I am mainly interested in the $3$-dimensional case. It is a well-known fact, following from the work of E. E. Moise and R. H. Bing in the 1950s, that every $3$-dimensional topological manifold (with ...
4 votes
0 answers
222 views

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 ...
4 votes
0 answers
119 views

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 $...
8 votes
2 answers
438 views

Quantitative word problem for 3-manifold groups

The word problem for 3-manifold groups is solvable: given a based loop $\gamma$ in $M^3$ there is an algorithm to decide whether $\gamma$ bounds a disk. What kinds of quantitative results are known ...
29 votes
3 answers
3k views

What part is left unsolved in the Unknotting problem? (after results of Bar-Natan, Khovanov, Kronheimer and Mrowka)

Kronheimer and Mrowka showed that the Khovanov homology detects the unknot. Bar-Natan showed a program to compute the Khovanov homology fast: there was no rigorous complexity analysis of the algorithm,...
1 vote
0 answers
116 views

(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 ...
5 votes
3 answers
497 views

Triangulations of 3-manifolds in Regina and SnapPy

I have been doing some statistical studies on small 3-manifolds, and I note that one can produce larg-ish censuses of triangulations in Regina. Now, the Regina documentation tells us how to convert a ...
27 votes
2 answers
1k views

Is being simply connected very rare?

Essentially, my question is how strong a restriction it is to be simply connected. Here is a way of making this precise: Let's say we want to count simplicial complexes (of dimension 2, though that ...
6 votes
3 answers
1k views

Algorithm for computing the Arf invariant of a knot

According to "The knot book", by Colin Adams, two knots are pass equivalent if they are related by a finite sequence of pass-moves. Moreover every knot is pass-equivalent to either the unknot or the ...
1 vote
0 answers
97 views

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 ...
2 votes
1 answer
53 views

Uniform closure of a neighbourhood complex in the tritetragonal tiling

Consider a neighbourhood complex of eight vertices (red) with vertex configuration $(3.4)^3$ which gives rise to the tritetragonal tiling of the hyperbolic plane: Not knowing if this complex can be ...
3 votes
2 answers
165 views

Looking through a bunch of links for unlinks?

I am looking for a bit of orientation with regards to computational topology resources, as I am personally totally ignorant on the subject. I have lots of different links in $S^3$ (hundreds of ...
11 votes
2 answers
678 views

Software for computing Thurston's unit ball

Is there any software which can be used for computing Thurston's unit ball (for second homology of 3-manifolds) of link complements? In particular can I do that with SnapPy? PS: even a table for ...
7 votes
1 answer
222 views

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 ...
0 votes
2 answers
249 views

Representations of modular lattices, extension to cellular sheaves

There are various "representation theorems" for lattices such as Birkhoff's Representation Theorem that states that every finite distributive lattice is isomorphic to a quasi-sublattice of the lattice ...
9 votes
1 answer
260 views

Smooth Morse function from Forman's discrete Morse function

Let $M$ be a smooth manifold and $K$ a triangulation of $M$, so $K$ is a regular CW-complex and in particular a simplicial complex. Assume that $M$ is compact so $K$ is finite. Let $f\colon K \to \...
2 votes
1 answer
180 views

On the entries of a matrix representation for a boundary operator of a persistence module

In equation 6 of Computing Persistent Homology (page 8), the authors put forward the following identity: $$\deg \hat{e_i}+\deg M_k (i,j)=\deg e_j$$ Where $\hat{e_i}$ and $e_j$ are elements of ...
2 votes
0 answers
181 views

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 ...
5 votes
2 answers
334 views

Is there an upper bound on the number of points in point cloud for which we compute the persistent homology?

I am interested in the topic of persistent homology (topological data analysis). According to what I read, there is some roadblock in the analysis of "big data" using persistent homology as it is ...
0 votes
1 answer
377 views

Clarification of "death event" in persistent homology

Before I ask my question let me clarify some notation: $f^{i,j}_r$, where $i < j$, refers to the inclusion map $f: H_r(X_i) \hookrightarrow H_r(X_j)$. $X_i$ and $X_j$ are subcomplexes of a filtered ...
24 votes
3 answers
2k views

Persistent homology of Gaussian fields in Euclidean space

If you generate points in $\mathbb R^n$ via a process that respects a Gaussian normal distribution, then compute the persistent homology / barcodes, to my eye something fairly regular seems to be ...
35 votes
1 answer
3k views

Why is persistent cohomology so much faster than persistent homology

I refer to this paper: de Silva, Vin; Morozov, Dmitriy; Vejdemo-Johansson, Mikael. Dualities in persistent (co)homology. Inverse Problems 27 (2011), no. 12, 124003, 17 pp. (Journal link, arXiv link). ...
7 votes
0 answers
176 views

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 ...
40 votes
5 answers
3k views

Reference on Persistent Homology

I will be teaching a course on algebraic topology for MSc students and this semester, unlike previous ones where I used to begin with the fundamental group, I would like to start with ideas of ...
3 votes
0 answers
122 views

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 ...
3 votes
0 answers
172 views

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 ...
18 votes
3 answers
622 views

Vietoris-Rips complex homology of a higher degree than the ambient dimension

Assuming we have a set of points $X=\{x_1,..,x_n\}$, all in $\mathbb{R}^d$, and construct the Vietoris-Rips-Complex $V_\epsilon (X)$ for some distance parameter $\epsilon > 0$. Is it possible to ...
7 votes
1 answer
1k views

Products, coproducts and equalizers in category of lattices

Background: Let $\mathbf{Lat}$ be the 2-category of lattices which can be viewed as a subcategory of the 2-cateogry of posets $\mathbf{Pos}$, that is, objects in $\mathbf{Pos}$ that have all finite ...
11 votes
0 answers
154 views

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 ...
3 votes
1 answer
141 views

An algorithm to tell if two cut systems are handle slide equivalent?

Let $\Sigma$ be a genus $g$ closed orientable surface and let $\alpha = \{ \alpha_1,...,\alpha_g\}$ and $\beta = \{ \beta_1 ,..., \beta_g \}$ be two cut systems (i.e. nonintersecting homologically ...
0 votes
1 answer
317 views

QUBO formulation of a discrete-variable Genetic Algorithm optimization problem

I am facing a non-linear, discrete optimization problem, which I can formulate in this abstract manner: I have a certain non-analytic real-valued function $f$ depending on a set of parameters $ \theta\...