Place topological defects in a 2D XY model. The winding number counts how many times the director field rotates around a defect. The total topological charge is conserved.
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Winding: ∮dθ/dl = 2πn
Core energy ~ log(R/a)
Charge conservation: defects created in pairs
Arrows show director orientation θ(r) = Σᵢ nᵢ · arg(r−rᵢ). Color encodes angle. Defects with |charge|=½ are nematic; |charge|=1 are polar vortices.