Timeline for Are small knots generic?
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Jun 5, 2013 at 7:53 | comment | added | Sam Nead | @unknown: even if Misha's objections somehow magically disappeared, your suggestion is unworkable. The waiting time is too long. | |
| Jun 5, 2013 at 7:51 | answer | added | Sam Nead | timeline score: 3 | |
| May 22, 2013 at 18:18 | comment | added | Misha | ...groups are hyperbolic! And in almost all of these group-theoretic models (except the oldest one), one does not even fix the number of generators (which would correspond to fixed genus/braid number). | |
| May 22, 2013 at 18:15 | comment | added | Misha | ... the feeling that "random" closed 3-manifold is hyperbolic. This is true in some models (where you fix genus and do a random walk on the mapping class group). You could do the same for knots if you work with braid position of a knot (fix the braid number and do random walk in the braid group). However, it goes against the "physical" appearance of knots, since these guys do not like to show up in the braid position! (Same with plat position). From the geometric group theory viewpoint, this is all, again, very strange, since in all natural model of randomness, random finitely-presented ... | |
| May 22, 2013 at 18:11 | comment | added | Misha | @unknown: It is easier said than done. You would have to come up with a simple combinatorial criterion which would allow you to say that a knot diagram gives rise to a hyperbolic knot. Of course, you can say that Haken's algorithm would detect essential tori, but this cannot be done in a "simple" combinatorial fashion (otherwise, say, knottedness probably would be in P). If you try to implement "pick a random knot" is that a "random knot diagram" will not be hyperbolic! (Which was your first comment.) Same problem with prime knots. In some sense, it is very counter-intuitive and goes against.. | |
| May 22, 2013 at 15:55 | comment | added | John Pardon | @Misha and Julien: why not just condition on the random knot being hyperbolic (or prime, etc.)? [concretely: pick random knots until you get a hyperbolic/prime one]. | |
| May 22, 2013 at 7:59 | comment | added | Julien Marché | I do not believe either in a nice random model producing only hyperbolic knots... that's why I will try to see if Snappy can detect smallness. After all, one can see in this way that most knots are non-alternating. | |
| May 22, 2013 at 4:18 | answer | added | Makoto Ozawa | timeline score: 3 | |
| May 21, 2013 at 16:39 | comment | added | Misha | I do not see how one can build "primality" condition into any feasible random model for knots; same for hyperbolicity. I do not know about Snappy knot census. In any case, that would cover only a finite number of knots and is likely to be misleading. | |
| May 21, 2013 at 15:56 | comment | added | John Pardon | For prime knots, I believe this result (ams.org/mathscinet-getitem?mr=1304395) implies that (for a certain sense of random knots) most are satellite knots (have an incompressible torus). So I would guess you also want to restrict to hyperbolic knots. Even in that case my guess would be that with lots of crossings you get lots of incompressible surfaces generically. | |
| May 21, 2013 at 13:12 | comment | added | Julien Marché | I agree, this is a "cheap" question, but I am more interested in a feeling than in a proof. In particular, do you know if it is implemented in snappy so that I can check for small crossing number? Thanks also for your remark, I should add that the knot K is prime... | |
| May 21, 2013 at 12:19 | comment | added | Misha | Questions like this are easy to ask but very hard to answer with the present technology, since crossing number of a knot $K$ is defined as the minimal crossing number of knot diagrams for $K$. A more realistic question would be to consider sets of all knot diagrams with $\le n$ crossings. I do not know the answer in this setting. But I do know that if you consider knot diagrams with edges on the rectangular grid, then with probability approaching $1$ (as $n\to \infty$), $K=K' \# T$, where $T$ is the trefoil and $K'$ is nontrivial, so the answer is strongly negative. | |
| May 21, 2013 at 8:00 | history | asked | Julien Marché | CC BY-SA 3.0 |