Timeline for Smoothing tame topological knots, from an analytic perspective
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Apr 22, 2022 at 23:39 | comment | added | Ryan Budney | @Echo: okay I think I see how to assemble an argument using your sketch. You really have to hit the problem over the head with a lot of 3-manifold theory. But I think there is an argument there. Unfortunately it's not the kind of organic argument I was hoping for. | |
| Apr 22, 2022 at 23:37 | history | edited | Ryan Budney | CC BY-SA 4.0 |
flesh out the strategy described by Echo in the comments.
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| Apr 22, 2022 at 7:54 | comment | added | user473423 | Right, I overlooked that. So let's say you assume additionally, that your knot admits a diagram with finitely many crossings (maybe that's automatically true), but then I guess you can modify the usual process of constructing a Seifert surface in a way that you get a smooth surface outside an arbitrary small neighbourhood U of the knot. You can iterate this process to get ever closer to the knot and at the same time make the surfaces agree outside a somewhat bigger neighbourhood. That menas, they will converge to a Seifert surface which is smooth in the interior. | |
| Apr 22, 2022 at 7:26 | comment | added | Ryan Budney | Certainly there are smoothing processes that involve convolution, but I do not see one available in this circumstance. One of the main problems with convolution-type arguments is they tend to not preserve functions being embeddings. | |
| Apr 22, 2022 at 7:18 | comment | added | user473423 | You can smoothen a given Seifert surface by convolution. If you make the function, you convolve with, dependent on the distance to the boundary, you get the result you want. | |
| Apr 22, 2022 at 5:13 | history | asked | Ryan Budney | CC BY-SA 4.0 |