Timeline for What constant ensures hyperbolicity of Dehn surgery?
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Feb 21, 2014 at 16:51 | comment | added | Ilya Kofman | We showed a universal bound for the one-cusped case (i.e. knots) in "The 500 simplest hyperbolic knots," arXiv:1307.4439. | |
| Jan 6, 2014 at 10:43 | vote | accept | shestipalov | ||
| Dec 31, 2013 at 13:56 | vote | accept | shestipalov | ||
| Jan 6, 2014 at 10:43 | |||||
| Dec 31, 2013 at 11:30 | comment | added | Bruno Martelli | I don't think you can recover a bound only from the volume $V$, you need to know the manifold. The cusp shapes tell you easily which curves have norm bigger than 6. With SnapPy simply use the function get_cusp_shape(). | |
| Dec 30, 2013 at 12:51 | comment | added | shestipalov | So perhaps a better question is: given a hyperbolic link of $n$ components which has volume $V$, can we explicitly find a constant $c(n,V)$ (so depending on the concrete manifold only minimally), such that filling $n-1$ components with slopes $p_i/q_i$ s.t. the norm of $(p_i,q_i)$ is bigger than this constant, we get a hyperbolic manifold with one cusp? Sorry for this long comment-question. | |
| Dec 30, 2013 at 12:50 | comment | added | shestipalov | My problem is (before I do learn hyperbolic geometry) that I have infinite families of knots, and all I can say at the moment, that infinitely many of them are hyperbolic. But what I would really like to say is given that parameters (that come from slopes) are bigger than a concrete constant, then the knots are hyperbolic. SnapPy can just verify this for a finite number of them. | |
| Dec 30, 2013 at 12:50 | comment | added | shestipalov | Thanks a lot for your answer! Somehow I got an incorrect idea that denominators were more important - thanks for clarifying that. Yes, I have SnapPy and seen references to Thurston's and Agol-Lackenby's theorems, but unfortunately have not yet had time to study this properly. So unfortunately I can't really understand what you are saying about the cusp shapes - that's something for me to learn, thanks for pointing this out. | |
| Dec 29, 2013 at 13:23 | history | edited | Bruno Martelli | CC BY-SA 3.0 |
added 10 characters in body
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| Dec 29, 2013 at 12:55 | history | answered | Bruno Martelli | CC BY-SA 3.0 |