# Determine the Winding Numbers of the Chinese Unicom Symbol

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:

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!

• Crossing the curve (at a point where the curve doesn't intersect itself) changes the winding number by $\pm 1$. So the winding numbers of $z_1$ and $z_3$ must differ by $1$, not $4$. Mar 18, 2015 at 6:54
• Okay, I agree with you that $W(z_3) \ne -4$. But when I trace the curve around $z_3$ I keep getting $W(z_3)=-2$ now. This one is tricky. Mar 18, 2015 at 7:12
• $W(z_3)$ is definitely $-1$. But if you want to discuss it, you're going to have to label your intersection points. Mar 18, 2015 at 10:23

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, $\mathrm{W}(z_5)=-1$.

• Great explanation! I didn't know it was this easy. I thought I had to trace the curve and keep track of the direction of every boundary around a point to see how the Winding number changes... This is a much simpler approach. Thank you! Mar 18, 2015 at 10:44
• In essence, this method does just that. By noting the direction of the curve at each intersection with a path, we are keeping track of the direction at each boundary along the path.
– robjohn
Mar 18, 2015 at 15:40

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.