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Results tagged with computational-topology
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user 1650
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.
39
votes
Accepted
What part is left unsolved in the Unknotting problem? (after results of Bar-Natan, Khovanov,...
EDIT: Marc Lackenby has just announced a quasi-polynomial time algorithm. That is, given an $n$—crossing diagram, the algorithm takes $n^{O(\log(n))}$ time to either find a spanning disk (proving the …
- 28.1k
3
votes
Accepted
An algorithm to tell if two cut systems are handle slide equivalent?
Yes, here is a sketch of one such proof. Let $V = V_\alpha$ be the genus $g$ handlebody determined by $\alpha$. The curves $\beta$ now give conjugacy classes in $\pi_1(V)$. Any one of the $\beta_i$ …
- 28.1k
3
votes
Accepted
Looking through a bunch of links for unlinks?
It will of course depend on where your examples are coming from. But here are some lightweight approaches.
Randomize the triangulation of the link complement a few times and then simplify. Do you g …
- 28.1k
10
votes
Accepted
Knot Diffie–Hellman
Here I assume that by “addition” of knots you mean the usual connect sum, as defined here. With that said, I think you correctly ask the relevant question: “Is factoring knots difficult?”
In favour o …
- 28.1k
9
votes
Accepted
Software for computing Thurston's unit ball
"Better late than never." Stephan Tillmann and William Worden have produced the software package tnorm. This can be found here:
https://pypi.org/project/tnorm/
The software should be able to deal wi …
- 28.1k
13
votes
Accepted
Is there an algorithm for the genus of a knot?
Jaco and Oertel's paper An algorithm to decide if a three-manifold is a Haken manifold [1984], plus a bit of work, gives a doubly exponential time algorithm to compute the Seifert genus. (In practice …
- 28.1k
8
votes
Accepted
Properties a triangulation must have in order to describe a manifold
From the comments:
Suppose that $T$ is a triangulation. If all vertex links are PL $(n-1)$-dimensional spheres then the realisation space of $T$ is a PL manifold and thus a topological manifold. (In …