Timeline for Computing self-intersections with complex analysis
Current License: CC BY-SA 4.0
21 events
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| May 21, 2020 at 7:06 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Apr 21, 2020 at 6:53 | history | edited | YCor | CC BY-SA 4.0 |
removed capitals from title (the question was bumped anyway)
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| Apr 20, 2020 at 18:32 | answer | added | Claudio Rea | timeline score: 2 | |
| Feb 10, 2018 at 5:17 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Jan 11, 2018 at 4:28 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Dec 12, 2017 at 3:47 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Nov 12, 2017 at 9:26 | comment | added | lcv | I would write it as a triple integral. Two for $z\in C$ and $w\in C$ and a third one to enforce “the delta” $z=w$. Then one should be able to transform it into a double integral... On second thought: .. well something like this. | |
| Nov 12, 2017 at 3:45 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| S Oct 13, 2017 at 3:40 | history | suggested | jeq | CC BY-SA 3.0 |
Copied image to imgur.com, as it was not being displayed because of the new https rule.
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| Oct 13, 2017 at 1:47 | review | Suggested edits | |||
| S Oct 13, 2017 at 3:40 | |||||
| Apr 17, 2015 at 22:30 | comment | added | Michael | Gauss linking number formula may be related. One "only" has to figure out how to produce a displacement $C'$ of $C$ so that their linking number would be related to $C$'s self-intersection number... | |
| Jun 21, 2014 at 11:39 | comment | added | PVAL | @TomGoodwillie Whitney gives an explicit definiton of "self-intersection number" in his paper "The self-intersections of a smooth n-manifold in 2n-space". For n even this is an integer invariant. For n odd (i.e. n=1) , this is only a mod $2$ invariant, and this can be seen by picking the opposite orientation on your manifold when calculating the intersection number. | |
| Jun 21, 2014 at 8:51 | answer | added | Andreas Rüdinger | timeline score: -1 | |
| S Aug 8, 2013 at 4:36 | history | suggested | Michael Albanese | CC BY-SA 3.0 |
Fixed up the two displayed equations.
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| Aug 8, 2013 at 2:42 | review | Suggested edits | |||
| S Aug 8, 2013 at 4:36 | |||||
| Mar 13, 2012 at 0:47 | comment | added | Tom Goodwillie | If the curve is immersed, and if its self-intersections occur transversely, then the number of these is of the opposite parity from the winding number. I don't see a way of specifying the multiplicity of a self-intersections as an integer, as opposed to a mod $2$ integer. | |
| Mar 12, 2012 at 19:15 | history | edited | Misha |
edited tags
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| Mar 11, 2012 at 6:15 | comment | added | Sungjin Kim | I think the winding number has no relation to self-intersections even if the curve is not completely intersecting itself. Consider a curve winds origin 2 times, with the first time, following unit circle, and second time, following hypocycloid(you can make the inner circle arbitrarily small so that the curve intersects the unit circle arbitrarily many times). Maybe we can give the lower bound of self-intersection number. | |
| Mar 11, 2012 at 2:50 | comment | added | john mangual | i believe Arnold showed the self-intersections + self-tangencies is conserved or something like that. To only count self-intersections seems harder -- try telling apart the cardioid (with a loop) and a trefoil. Maybe the combinatorial answer involves $f:\mathbb{C} \to \mathbb{N}$ by $f(z)=$ winding number of $C$ around $z$. | |
| Mar 11, 2012 at 0:04 | comment | added | user10290 | If you know that your path is closed and does not completely intersect itself, like a circle which just keeps winding around itself, then it seems to me that the winding number minus one will count the number of self-intersections or self-tangencies since every time it winds it must pass through itself or touch itself tangentially. This might just work for the examples I am visualizing and the ones in your nice picture. | |
| Mar 10, 2012 at 22:49 | history | asked | john mangual | CC BY-SA 3.0 |