TITLE: Can any sequence of consecutive integers be realized as winding numbers?
QUESTION [8 upvotes]: 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 $C$?

REPLY [21 votes]: 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.