Winding Number & Topological Charge

How many times does a field wind around a defect?
n = 0
W = (1/2π) ∮ dθ
Charge n +1
Field Type
Show loop
The winding number counts how many times a vector field rotates as you traverse a closed loop. Topological defects (vortices, skyrmions, monopoles) carry integer charges that cannot be removed by smooth deformations — they are topologically protected. The winding number is a topological invariant: homotopy class of maps S¹→S¹, classified by π₁(S¹) = ℤ.