Timeline for Can we determine which monodromy of surface gives a fibered knot?
Current License: CC BY-SA 4.0
12 events
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| S Aug 22, 2021 at 22:05 | history | suggested | Henry | CC BY-SA 4.0 |
corrected a typo
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| Aug 22, 2021 at 21:11 | review | Suggested edits | |||
| S Aug 22, 2021 at 22:05 | |||||
| May 31, 2012 at 7:24 | comment | added | Bruno Martelli | @Lee: Thank you, the argument is clear and beautiful. I have corrected a typo in your text (replaced S with M). | |
| May 31, 2012 at 7:21 | history | edited | Bruno Martelli | CC BY-SA 3.0 |
typo: replaced S with M
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| May 31, 2012 at 2:53 | history | edited | Lee Mosher | CC BY-SA 3.0 |
added 36 characters in body
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| May 31, 2012 at 2:48 | history | edited | Lee Mosher | CC BY-SA 3.0 |
added 2166 characters in body; added 24 characters in body
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| May 30, 2012 at 19:00 | comment | added | Lee Mosher | @Bruno: the longitude is the generator of the kernel of $H_1(\partial M;Z) \to H_1(M;Z)$. The degeneracy slope is defined under the assumption that the mapping class is pseudo-Anosov, so in $M$ there is an essential lamination $\Lambda$ which is the suspension of the stable measured lamination, and there is an embedded annulus $A \subset M$ which intersects $\Lambda$ in one component of $\partial A$ and intersects $\partial M$ in the other component of $\partial A$ which by definition is the degeneracy slope. | |
| May 30, 2012 at 15:37 | comment | added | Bruno Martelli | @Ian, I can see only one intrinsecally well-defined curve in the torus boundary, the boundary of the surface (is this what you call the longitude?). How can you tell among the infinitely many slopes intersecting it once, which all produce homology spheres (assuming H_1(M)=Z), those that give you S^3? | |
| May 29, 2012 at 21:12 | comment | added | Ian Agol | @ Igor: 3, since the slope has to have intersection number $\leq 1$ with both the longitude and degeneracy slope, which can therefore be at most 3 curves. | |
| May 29, 2012 at 18:26 | comment | added | Igor Rivin | How many is "finite"? Is there nan effective bound given OP's data (the monodromy), with the homology constraint? | |
| May 29, 2012 at 14:36 | comment | added | Ian Agol | I think you want to do Dehn filling along a slope which is either the loop generating the monodromy on the boundary, or a curve intersecting it once. But there are still only finitely many choices of slope by restricting to Dehn fillings which are homology spheres, since the slope must also intersect the longitude just once. | |
| May 29, 2012 at 12:50 | history | answered | Lee Mosher | CC BY-SA 3.0 |