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