Timeline for Is there a sensible notion of a winding number of a closed curve in $\mathbb{R}^n$, $n\geq 3$, with respect to a point not lying on it?
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Apr 11, 2018 at 18:03 | vote | accept | M.G. | ||
| Apr 10, 2018 at 15:03 | comment | added | M.G. | @JosephO'Rourke: thanks, I have seen your question and even posted an answer to it just now :-) However, the setting of your question is different in an essential way. | |
| Apr 10, 2018 at 14:56 | comment | added | Najib Idrissi | @IgorRivin Why do you want the curves to be isotopic? The winding number of curves in the plane is a homotopy invariant. Moreover "isotopic", to my knowledge, refers to embedded submanifolds. Embedded curves in $\mathbb{R}^2$ are somewhat pointless (and the winding number is either zero or one). | |
| Apr 10, 2018 at 14:43 | comment | added | Joseph O'Rourke | See also the earlier question, Generalization of winding number to higher dimensions. | |
| Apr 10, 2018 at 13:31 | history | edited | M.G. |
per comments added soft-question tag
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| Apr 10, 2018 at 3:28 | comment | added | Anthony Carapetis | Well, I guess it depends what properties of the winding number you're trying to generalize; such a soft question perhaps should not be answered with a definite yes or no. To me the ambient isotopy class doesn't seem to be a good answer to this question, especially given its insensitivity to any "vantage point" $p$. | |
| Apr 10, 2018 at 3:12 | comment | added | Igor Rivin | @AnthonyCarapetis being homotopic is irrelevant here - you want the curves to be isotopic, in which case your statement is false, as proved by the existence of the subject of "knot theory". | |
| Apr 10, 2018 at 3:12 | answer | added | Wlod AA | timeline score: 7 | |
| Apr 10, 2018 at 3:01 | comment | added | Anthony Carapetis | yes, this makes more sense - see Piotr's answer re linking numbers. | |
| Apr 10, 2018 at 2:58 | comment | added | M.G. | @AnthonyCarapetis: that's a really good point. At the same time it raises the question if one might be able to speak of a "winding number" of a curve with respect to some proper obstruction of codimension $2$ instead of a pt. | |
| Apr 10, 2018 at 2:57 | answer | added | Piotr Hajlasz | timeline score: 13 | |
| Apr 10, 2018 at 2:46 | comment | added | Anthony Carapetis | Any two closed curves not touching $p$ are homotopic in $\mathbb R^3 \setminus \{p\},$ so I don't see how a useful definition is possible. | |
| Apr 10, 2018 at 2:18 | history | asked | M.G. | CC BY-SA 3.0 |