Timeline for Fundamental problems whose solution seems completely out of reach
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Jul 3, 2012 at 11:21 | comment | added | Andrew | The only one invariant of knots which seems to be complete and many people beleve in it is Vasilliev' invariants, discovered by russian mathematician Vasilliev in 90-th. It was based on the ideas of singularity theory. M. Kontsevich also took a hand on them, constructed Kontsevich integral, and the whole question "redused" to combinatorics. But... My idea is that knot "nature" is still not known, I mean, it seems that it should inspire us to construct new mathematical tool. In this sence, due to Poincare, it is a good question. Sorry for my English. It is not my native language. | |
| Jul 3, 2012 at 0:49 | comment | added | Douglas Zare | I still don't know which problem is being suggested, but I think that with all of the progress on knot theory and geometrization, it would be very strange to say, "The solution seems completely out of reach." | |
| Jul 2, 2012 at 20:50 | comment | added | HJRW | That last comment was addressed to Andrew. | |
| Jul 2, 2012 at 20:49 | comment | added | HJRW | Yes, but you didn't mention anything about the complexity of the algorithm in your answer. When you say 'Given an arbitrary knot, you cannot, in general, determine its isotopy class', I presume you mean a knot in an arbitrary 3-manifold? If that's not known to be decidable, then I agree it would be a good answer to this question. | |
| Jul 2, 2012 at 20:10 | comment | added | Douglas Zare | Classifying knots doesn't seem to belong on this list at all. | |
| Jul 2, 2012 at 19:44 | comment | added | Andrew | That's right. But this algorithm depends on knot crossings so is hard to compute. And it doesn't close the classification problem. Given an arbitrary knot, you cannot, in general,determine its isotopy class neither by this algorithm nor by other invariants. P.S. As I know, Haken didn't finish the work on this algorithm. It was done by russian mathematician S.V.Matveev. | |
| Jul 2, 2012 at 10:13 | comment | added | HJRW | I'm confused. Haken's algorithm determines whether two knot-complements are homeomorphic and hence, by the Gordon--Luecke Theorem, whether the two knots are isotopic (modulo orientation, admittedly). | |
| Jul 2, 2012 at 5:50 | comment | added | Douglas Zare | It's still not clear to me what question you are asking. | |
| Jul 1, 2012 at 20:26 | comment | added | Andrew | I mean a classification of knots under the isotopy. Just formulated it in physical setting to show its simple nature. A physical type problem is also unsolved. There are some development here, uncluding algorithms, depending on the thickness of a knot. | |
| Jul 1, 2012 at 19:20 | comment | added | alvarezpaiva | Do you mean the study of knot types when the length and thickness of the knot must be kept constant during the isotopy? Any reference for this problem or at least for a hint of it? | |
| Jul 1, 2012 at 18:56 | history | answered | Andrew | CC BY-SA 3.0 |