 Solution

Homeotropic anchoring means that the nematic director is locally normal to the particle surface, hence radial on a sphere. Far away, the uniform nematic liquid crystal has a fixed director ${\mathbf{n}}_{\infty }$, with $\mathbf{n}$ and $-\mathbf{n}$ physically equivalent.

One plausible configuration is a quadrupolar Saturn-ring defect: the director is radial near the particle, bends into ${\mathbf{n}}_{\infty }$ far away, and has a disclination loop just outside the particle's equator, in the plane perpendicular to ${\mathbf{n}}_{\infty }$. A meridional section cuts the loop at two points.

The defect compensates the topological distortion introduced by the radial surface boundary condition. A dipolar configuration with a nearby hyperbolic point defect is another possible texture. Which is stable depends on particle size, anchoring strength, elastic constants and external confinement; the boundary conditions alone do not uniquely select one.^[ref]