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It is possible to find the winding number of a path $C \subset \mathbb{C}$ using complex analysis:

$$n = \oint_C\frac{dz}{z}.$$ You can also count the number of roots of $f(z) = 0$ inside a close curved $C$:

$$n = \oint_C dz\,\frac{f'(z)}{f(z)}.$$

How do we count the number of self-intersections or self-tangencies of $C$ with itself?

$$\#[C \cap C] = \oint \frac{dz}{z}\frac{dw}{w} f(z,w)?$$ Self-intersection is a global event depending on two points - probably it is some kind of double-integral.

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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. –  Erin K Carmody Mar 11 '12 at 0:04
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 $z$. –  john mangual Mar 11 '12 at 2:50
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. –  i707107 Mar 11 '12 at 6:15
Maybe you can write the number of self intersections cohomologically, and then use de Rham's theorem to give you can integral representation. –  Steven Gubkin Mar 12 '12 at 21:52
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. –  Tom Goodwillie Mar 13 '12 at 0:47
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