  In this paper we study supersymmetric co-dimension 2 and 4 defects in the
compactification of the 6d $(2,0)$ theory of type $A_{N-1}$ on a 3-manifold
$M$. The so-called 3d-3d correspondence is a relation between complexified
Chern-Simons theory (with gauge group $SL(N, \mathbb{C})$) on $M$ and a 3d
$\mathcal{N}=2$ theory $T_{N}[M]$. We establish a dictionary for this
correspondence in the presence of supersymmetric defects, which are knots/links
inside the 3-manifold. Our study employs a number of different methods:
state-integral models for complex Chern-Simons theory, cluster algebra
techniques, domain wall theory $T[SU(N)]$, 5d $\mathcal{N}=2$ SYM, and also
supergravity analysis through holography. These methods are complementary and
we find agreement between them. In some cases the results lead to highly
non-trivial predictions on the partition function. Our discussion includes a
general expression for the cluster partition function, in particular for
non-maximal punctures and $N>2$. We also highlight the non-Abelian description
of the 3d $\mathcal{N}=2$ $T_N[M]$ theory with defect included, as well as its
Higgsing prescription and the resulting `refinement' in complex CS theory. This
paper is a companion to our shorter paper arXiv:1510.03884, which summarizes
our main results.
