  The homotopy theory of topological defects is a powerful tool for organizing
and unifying many ideas across a broad range of physical systems. Recently,
experimental progress has been made in controlling and measuring colloidal
inclusions in liquid crystalline phases. The topological structure of these
systems is quite rich but, at the same time, subtle. Motivated by experiment
and the power of topological reasoning, we review and expound upon the
classification of defects in uniaxial nematic liquid crystals. Particular
attention is paid to the ambiguities that arise in these systems, which have no
counterpart in the much-storied XY model or the Heisenberg ferromagnet.
