Half-integer and integer winding number defects in liquid crystals
In nematic liquid crystals, molecules align along a director field n(r) — a headless vector (n ≡ −n). Defects are classified by their winding number (charge) k:
The ±½ defects are topologically stable because the fundamental group π₁(RP²) = Z₂. A full rotation (k=1) can be unwound, but a half-rotation (k=½) cannot.
Defects of opposite charge attract and annihilate. Same-sign defects repel. This drives self-organization in active nematics (cytoskeleton, bacteria colonies).