  We derive two geometric approaches to categorification of quantum invariants
of links associated to an arbitrary compact simple Lie group $^L{G}$. In part
I, we describe the first approach, based on an equivariant derived category of
coherent sheaves on ${\cal X}$, the moduli space of singular $G$-monopoles,
where $G$ is related to $^LG$ by Langlands duality. In part II, we describe the
second approach, based on the derived category of a Fukaya-Seidel category of a
Calabi-Yau $Y$ with potential $W$. The two approaches are related by a version
of mirror symmetry, which plays a crucial role in the story. In part III, we
explain the string theory origin of these results, and the relation to an
approach due to Witten.
