Determine the Winding Numbers of the Chinese Unicom Symbol

Question

I'm practicing with Winding Numbers, and encountered an interesting example. You might be familiar with this liantong symbol, the logo of China Unicom:

Suppose we make this into a fully closed and connected curve, and try to determine the Winding Numbers of the various points in the symbol. For instance:

Find the winding numbers of the closed curve shown below at $z_1,z_2,z_3,z_4,z_5$

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It seems to me that for each $z$, the winding number $W(z)$ is:

1. $W(z_1)=0$ (since it is outside the curve)
2. $W(z_2)=1$ (since it falls to the left of the curve in one loop)
3. $W(z_3)=-2$ (since it falls to the right of the curve in two loops)
4. $W(z_4)=0$ (since it falls to the right and to the left of the curve twice each, cancelling out)
5. $W(z_5)=-1$ (since it falls to the right of the curve in one loop)

Would you agree with these winding numbers (and given reasoning)? Thank you for your help!

Answer 1

If a curve crosses your path from left to right, the winding number increases.

If a curve crosses your path from right to left, the winding number decreases.

Going from $z_1$ to $z_2$, the curve crosses our path from left to right. Therefore, $\mathrm{W}(z_2)=1$.

Going from $z_1$ to $z_3$, the curve crosses our path from right to left. Therefore, $\mathrm{W}(z_3)=-1$.

Going from $z_3$ to $z_4$, the curve crosses our path from left to right. Therefore, $\mathrm{W}(z_4)=0$.

Going from $z_4$ to $z_5$, the curve crosses our path from right to left. Therefore, $\mathrm{W}(z_5)=-1$.

Answer 2

Note that the curve can be deformed to a union of clockwise and counterclockwise circles, which makes it much easier to find the winding numbers.