For a closed plane curve $C$, define its sequence of winding numbers to be the sorted list of the winding numbers of each of the distinct regions of the plane demarcated by $C$. For example, this curve (if I've calculated correctly) has sequence $001111223 = 0^2 1^4 2^2 3$.

A winding number sequence must include $0$ for the unbounded region of the plane. I am wondering if there are any other restrictions:

Q. Can any winding sequence of consecutive integers that includes $0$ be realized by some curve $C$?

  • $\begingroup$ Maybe a silly question, but why do the numbers need to be consecutive? You can go around any loop more than once. (Perhaps you are just imposing consecutiveness for simplicity?) $\endgroup$ – Zach Teitler Dec 31 '17 at 23:39
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    $\begingroup$ @ZachTeitler: "the winding numbers for any two adjacent regions differ by exactly 1; the region with the larger winding number appears on the left side of the curve." This from Wikipedia. $\endgroup$ – Joseph O'Rourke Dec 31 '17 at 23:52
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    $\begingroup$ Ah, the curve is forbidden from retracing itself. Okay. $\endgroup$ – Zach Teitler Jan 1 '18 at 0:25

Isn't this easy by induction? Delete one of the largest numbers, say $m$, from your sequence, realize the remaining numbers, then in the realization pick any region with winding number $m-1$, and make an extra loop somewhere close to its boundary.

  • $\begingroup$ Simple, clean, and convincing! $\endgroup$ – Joseph O'Rourke Dec 31 '17 at 21:22
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    $\begingroup$ Also works for the signed version of the question by looping either clockwise or counter-. $\endgroup$ – Noam D. Elkies Dec 31 '17 at 21:40
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    $\begingroup$ Naively, how about for a higher-dimensional version, such as in mathoverflow.net/questions/259054/… ? $\endgroup$ – Zach Teitler Dec 31 '17 at 23:13
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    $\begingroup$ @ZachTeitler: Thanks, Zach, that's where I was heading next... Perhaps a future question. $\endgroup$ – Joseph O'Rourke Dec 31 '17 at 23:53
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    $\begingroup$ I don't see why that would require a different solution, you can just do the same thing. $\endgroup$ – domotorp Jan 1 '18 at 11:46

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