  We give a homotopy classification of the global defects in ordered media, and
explain it via the example of biaxial nematic liquid crystals, i.e., systems
where the order parameter space is the quotient of the $3$-sphere $S^3$ by the
quaternion group $Q$. As our mathematical model we consider continuous maps
from complements of spatial graphs to the space $S^3/Q$ modulo a certain
equivalence relation, and find that the equivalence classes are enumerated by
the six subgroups of $Q$. Through monodromy around meridional loops, the edges
of our spatial graphs are marked by conjugacy classes of $Q$; once we pass to
planar diagrams, these labels can be refined to elements of $Q$ associated to
each arc. The same classification scheme applies not only in the case of $Q$
but also to arbitrary groups.
