Winding Number & Topological Charge

How many times does a closed curve wind around a point? A topological invariant.

Vortex charge n +1

Loop radius 80

Loop center X 0

Loop center Y 0

Left: vector field of a topological vortex with charge n. Right: angle traversed along loop.
Winding number = (1/2π)∮ dφ — integer-valued, homotopy invariant. Changes only when loop crosses singularity.