Timeline for Knot theory without planar diagrams?
Current License: CC BY-SA 3.0
21 events
| when toggle format | what | by | license | comment | |
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| Mar 11, 2019 at 17:59 | answer | added | Michael | timeline score: 1 | |
| Mar 11, 2019 at 17:45 | history | edited | YCor |
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| Jan 27, 2013 at 7:28 | answer | added | Sammy Black | timeline score: 2 | |
| Aug 7, 2012 at 10:01 | answer | added | Daniel Moskovich | timeline score: 7 | |
| S Aug 7, 2012 at 9:01 | vote | accept | Marius Buliga | ||
| Aug 7, 2012 at 9:00 | vote | accept | Marius Buliga | ||
| S Aug 7, 2012 at 9:01 | |||||
| Aug 7, 2012 at 4:18 | comment | added | Misha | Haken also should be mentioned here since the only known algorithm for telling one knot from the other goes back to his theory of normal surfaces and has nothing to do with knot diagrams. Incidentally, distinguishing knots is the original raison d'être for the entire field of knot invariants and none of them so far can do it (for arbitrary knots). | |
| Aug 7, 2012 at 1:32 | comment | added | Charlie Frohman | I think most of three manifold topology is about doing knot theory without diagrams. Only asking for one paper seems out of proportion to the amount of literature dedicated to doing knot theory without diagrams. For instance, Dave Gabai's proof of property R, Gordan and Luecke's solution to the knot complement problem. Thurston's work on geometrization, the whole field of normal surface theory.... | |
| Aug 7, 2012 at 0:18 | comment | added | Ryan Budney | I think this is a good question, in that because there's such a relatively high volume of knot theory through the eyes of planar diagrams, it can easily give the impression that this is all there is. Frequently people will give surveys of knot theory and never mention Schubert, Seifert, Waldhausen or Thurston. I've been present in the audience at least three times for such surveys. There's of course two reasons for this: (1) new invariants originally couched in the language of planar diagrams and (2) this is a low-overhead approach to knot theory, so it's easier to draw in undergrads. | |
| Aug 6, 2012 at 23:57 | comment | added | Ryan Budney | Marius, yes of course you can compute the Alexander polynomial without using a projection -- frequently this is the most efficient way to do it. The Alexander polynomial of torus knots are most easily computed using the Seifert fibering of the knot complement. This gives you a beautiful cell decomposition of it with one cell in dimensions 0 and 2, and two cells in dimension 1. | |
| Aug 6, 2012 at 23:54 | answer | added | Ryan Budney | timeline score: 11 | |
| Aug 6, 2012 at 18:56 | answer | added | Jim Conant | timeline score: 10 | |
| Aug 6, 2012 at 18:29 | answer | added | Igor Rivin | timeline score: 2 | |
| Aug 6, 2012 at 18:26 | comment | added | Marius Buliga | Well, you can define the Alexander polynomial by using the knot complement, but can you compute it without using a convenient projection which gives you a knot diagram to work with? I don't know and I would like to learn if this was done somewhere (and get a reference). | |
| Aug 6, 2012 at 18:24 | comment | added | Daniel Moskovich | What are "the obvious topological invariants"? Does e.g. concordance information count? (whether or not a knot is slice, for example) | |
| Aug 6, 2012 at 18:21 | comment | added | Daniel Moskovich | I think almost none of classical knot theory uses knot diagrams. So almost all of Burde-Zieschang, for example. This is a bit of a sad question for me... is this what things look like now; knots are just 3-dimensional representations of knot diagrams?? | |
| Aug 6, 2012 at 18:19 | comment | added | Qiaochu Yuan | @Marius: is the Alexander polynomial an obvious topological invariant defined from the complement of a knot? | |
| Aug 6, 2012 at 18:19 | answer | added | Qiaochu Yuan | timeline score: 2 | |
| Aug 6, 2012 at 18:10 | comment | added | Marius Buliga | This is a reference-request, for example for a paper concerning knot invariants which do not involve knot diagrams in order to define or compute them, say other than the obvious topological invariants defined from the complement of a knot. Could you please me indicate such a reference? Thanks! | |
| Aug 6, 2012 at 18:04 | comment | added | Igor Rivin | The answer is: YES. The question should be community wiki. | |
| Aug 6, 2012 at 17:53 | history | asked | Marius Buliga | CC BY-SA 3.0 |