  In fivebrane compactifications on 3-manifolds, we point out the importance of
all flat connections in the proper definition of the effective 3d N=2 theory.
The Lagrangians of some theories with the desired properties can be constructed
with the help of homological knot invariants that categorify colored Jones
polynomials. Higgsing the full 3d theories constructed this way recovers
theories found previously by Dimofte-Gaiotto-Gukov. We also consider the
cutting and gluing of 3-manifolds along smooth boundaries and the role played
by all flat connections in this operation.
