  The conventional topological description given by the fundamental group of
nematic order parameter does not adequately explain the entangled defect line
structures that have been observed in nematic colloids. We introduce a new
topological invariant, the self-linking number, that enables a complete
classification of entangled defect line structures in general nematics, even
without particles, and demonstrate our formalism using colloidal dimers, for
which entangled structures have been previously observed. We also unveil a
simple rewiring scheme for the orthogonal crossing of two -1/2 disclinations,
based on a tetrahedral rotation of two relevant disclination segments, that
allows us to predict possible nematic braids and calculate their self-linking
numbers.
