Find the winding number around $z=-i, z=-1, z=0$ in the following figure.
The purpose of this exercise is to complete a complex integral with singularities at the stated points. My attempt is that the winding number around $z=0$ is $1$, and that the winding numbers around $z=-i$ is zero, and $z=-1$ is $-1$. The reason it is zero for $z=-i$ is because the curve winds around $-i$ clockwise once, and then winds around it counterclockwise once. The reason it is $-1$ for $z=-1$ is because the curve winds around that point clockwise once.
My shakiest solution is the winding number for $z=-1$. For some reason, I have a competing thought that the winding number is in fact zero. In my head, I figure I'm free to "move" the knot around $z=-i$ and "unfold" it to get a figure that does not wind around $z=-1$ at all. Informally, I think of the curve as a rope where the intersections are actually a part of the rope lying on top of another part. I hope that is not too unclear.