= Solution
A <topological defect> occurs where the local order cannot be continued smoothly while retaining its surrounding texture. At a <nematic disclination> core, the director model fails and the orientational order may decrease.
For a planar texture $\mathbf n=(\cos\phi,\sin\phi)$,
$$
s=\frac1{2\pi}\oint d\phi.
$$
Because $\mathbf n\equiv-\mathbf n$, half-integer winding strengths are allowed. The original sketches use $\phi=s\alpha+\phi_0$ to show \b[$s=+1/2,-1/2,+1$], corresponding to director turns $+\pi,-\pi,+2\pi$. A radial $+1$ pattern and an azimuthally rotated version have the same winding.
These are planar labels. In unrestricted three dimensions an integer line texture may escape by tilting out of the plane; half-integer lines have a different classification. A planar disclination strength must not be identified automatically with a three-dimensional hedgehog point charge.
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