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We study nematic equilibria in an unbounded domain, with a two-dimensionalregular polygonal hole with $K$ edges, in a reduced Landau-de Gennes framework.This complements our previous work on the "interior problem" for nematicequilibria confined inside regular polygons (SIAM Journal on AppliedMathematics, 80(4):1678-1703, 2020). The two essential dimensionless modelparameters are $\lambda$--the ratio of the edge length of polygon hole to thenematic correlation length, and an additional degree of freedom,$\gamma^*$--the nematic director at infinity. In the $\lambda\to 0$ limit, thelimiting profile has two interior point defects outside a generic polygon hole,except for a triangle and a square. For a square hole, the limiting profile haseither no interior defects or two line defects depending on $\gamma^*$, and fora triangular hole, there is a unique interior point defect outside the hole. Inthe $\lambda\to\infty$ limit, there are at least $\binom{K}{2}$ stable states%differentiated by the location of two bend vertices and the multistability isenhanced by $\gamma^*$, compared to the interior problem. Our work offers newinsights into how to tune the existence, location, and dimensionality ofdefects.

Multistability for a Reduced Nematic Liquid Crystal Model in the Exterior of 2D Polygons