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3
votes
0answers
276 views

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?
16
votes
3answers
515 views

What is the state of the art for algorithmic knot simplification?

Question: Given a `hard' diagram of a knot, with over a hundred crossings, what is the best algorithm and software tool to simplify it? Will it also simplify virtual knot diagrams, tangle diagrams, ...
14
votes
2answers
538 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 ...
4
votes
1answer
332 views

Persistent homology of Markovian dynamical systems

Consider a dynamical system $(T,X)$ that admits a Markov partition $\mathcal{M}$ (e.g., an Anosov map), and consider the corresponding 0-1 transition matrix $A$. It is commonplace to study information ...
4
votes
1answer
277 views

Biggest ball included in an intersection of balls

I would like to prove that for any family of balls $\{B(c_i,r_i)\}_i \subset \mathbb{R}^d$ such that $\{c_1, \dots, c_n\} \subset \bigcap_i B(c_i,r_i) $ and $\forall i, r_i \geq 1$, there exists a ...
3
votes
1answer
158 views

Are there “geometrically nice” sets from which to construct coverings that admit “Vietoris-Rips like” approximations to the nerve?

It is well known that the nerve (or Čech complex) of a covering consisting of metric balls with a common fixed radius is nicely approximated by the Vietoris-Rips complex. Being a flag complex on its ...
3
votes
1answer
141 views

Triangulation of the surface determined by sampling two of its cross-sections

I have a data set that essentially looks like the picture below, i.e., it's given by sets of points in $\mathbb{R}^3$ that sample the cross-sections of a certain surface that in principle I do not ...
0
votes
2answers
608 views

Computational Topology Paper

Hi I am delving into the field of Computational Topology. I am aware of the books in this field, but could anybody tell me a nice relevant paper in this field which tackles a "typical" ...