TITLE: Determine the Winding Numbers of the Chinese Unicom Symbol
QUESTION [8 upvotes]: 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 $z_1,z_2,z_3,z_4,z_5$


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

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

REPLY [4 votes]: 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.