Timeline for Why should I care about Heegaard-Floer theory?
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Oct 9 at 13:33 | comment | added | HJRW | Understanding the computational complexity of unknot recognition is a major open problem. Breakthroughs have recently been made by Lackenby, but I don't think Heegaard--Floer is involved in his arguments. A reference showing that Heegaard--Floer provides "effective algorithms" for, well, anything that can't be computed by other methods, is needed to make this answer credible. | |
| Feb 12, 2018 at 6:31 | comment | added | dvitek | An update on the state of the art in computations in Heegaard-Floer homology: Ozsvath-Szabo's recent bordered matchings package seems to extend the range of effective computation of knot Floer homology to significantly larger knots -- my understanding is that stuff on the order of 30-40 crossings is now possible. | |
| Feb 19, 2012 at 1:56 | comment | added | Sean Tilson | Frequently new methods of computation gives new insight. This is certainly the case for homotopy groups of spheres. | |
| Feb 18, 2012 at 4:37 | comment | added | Mitya | Yes, but computing Heegaard Floer invariants is still not very efficient. I'm not sure about knot genus calculations, but tables of isotopy classes of knots were computed up to 16 crossings using classical methods. My understanding is that the algorithms for Heegaard Floer computations are not computationally practical for that number of crossings. Thus the fact that they detect the unknot (much weaker than separating isotopy classes) adds nothing computational to the picture. | |
| Feb 17, 2012 at 8:01 | comment | added | Mariano Suárez-Álvarez | Indeed! In a similar vein, algorithms for computing the homotopy groups of spheres have been known for ages. | |
| Feb 17, 2012 at 6:58 | history | answered | Michael Hutchings | CC BY-SA 3.0 |