  We investigate prototypical profiles of point defects in two dimensional
liquid crystals within the framework of Landau-de Gennes theory. Using boundary
conditions characteristic of defects of index $k/2$, we find a critical point
of the Landau-de Gennes energy that is characterised by a system of ordinary
differential equations. In the deep nematic regime, $b^2$ small, we prove that
this critical point is the unique global minimiser of the Landau-de Gennes
energy. We investigate in greater detail the regime of vanishing elastic
constant $L \to 0$, where we obtain three explicit point defect profiles,
including the global minimiser.
