Timeline for Knot and its embedded disk
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 9, 2019 at 21:15 | comment | added | Jonny Evans | The cone disc you're interested in (basically) the algebraic curve y^2=x^3 in C^2 when K is a trefoil knot (and has such an algebraic description whenever K is an iterated torus knot). This fact is used in some proofs of the identification of the complement with the homogeneous space of SL(2,R) for the trefoil. | |
| Aug 9, 2019 at 21:12 | comment | added | Jonny Evans | There seems to be a discussion with some links here: mathoverflow.net/questions/93942/… | |
| Aug 9, 2019 at 20:56 | comment | added | Diego Hernández Rodríguez | Thanks. Do you have a good reference for the trefoil? | |
| Aug 9, 2019 at 19:56 | comment | added | Jonny Evans | Incidentally, for the trefoil, the answer to 1 is $SL(2,R)/SL(2,Z)$, which is nice. For the figure 8 you can give a nice triangulation of the complement by two ideal hyperbolic tetrahedra: I think this is explained in Thurston's book on 3-manifolds. I think if you're interested in the cone disc then the answer to 2 should be the knot complement times R, but I'm not 100% sure I know what a "tubular neighborhood" is in that case. | |
| Aug 9, 2019 at 19:44 | comment | added | Jonny Evans | I see: this was not clear from what you wrote. | |
| Aug 9, 2019 at 18:37 | comment | added | Diego Hernández Rodríguez | Such a disk always exists. You can consider the cone the over the knot. I just dropped the smoothness condition. | |
| Aug 9, 2019 at 14:31 | comment | added | Jonny Evans | Such a D only exists if K is slice (by definition). Even then, I would imagine that the diffeomorphism type of the complement of a slice disc depends on which slice disc you pick. | |
| Aug 9, 2019 at 13:48 | history | edited | Diego Hernández Rodríguez | CC BY-SA 4.0 |
improved information
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| Aug 9, 2019 at 13:20 | review | First posts | |||
| Aug 9, 2019 at 13:51 | |||||
| Aug 9, 2019 at 13:15 | history | asked | Diego Hernández Rodríguez | CC BY-SA 4.0 |