  Topological gauge theories in four dimensions which admit surface operators
provide a natural framework for realizing homological knot invariants. Every
such theory leads to an action of the braid group on branes on the
corresponding moduli space. This action plays a key role in the construction of
homological knot invariants. We illustrate the general construction with
examples based on surface operators in N=2 and N=4 twisted gauge theories which
lead to a categorification of the Alexander polynomial, the equivariant knot
signature, and certain analogs of the Casson invariant.
  This paper is based on a lecture delivered at the International Congress on
Mathematical Physics 2006, Rio de Janeiro, and at the RTN Workshop 2006,
Napoli.
