  The recently conjectured knots-quivers correspondence relates gauge theoretic
invariants of a knot $K$ in the 3-sphere to representation theory of a quiver
$Q_{K}$ associated to the knot. In this paper we provide geometric and physical
contexts for this conjecture within the framework of the large $N$ duality of
Ooguri and Vafa, that relates knot invariants to counts of holomorphic curves
with boundary on $L_{K}$, the conormal Lagrangian of the knot in the resolved
conifold, and corresponding M-theory considerations. From the physics side, we
show that the quiver encodes a 3d ${\mathcal N}=2$ theory $T[Q_{K}]$ whose low
energy dynamics arises on the worldvolume of an M5 brane wrapping the knot
conormal and we match the (K-theoretic) vortex partition function of this
theory with the motivic generating series of $Q_{K}$. From the geometry side,
we argue that the spectrum of (generalized) holomorphic curves on $L_{K}$ is
generated by a finite set of basic disks. These disks correspond to the nodes
of the quiver $Q_{K}$ and the linking of their boundaries to the quiver arrows.
We extend this basic dictionary further and propose a detailed map between
quiver data and topological and geometric properties of the basic disks that
again leads to matching partition functions. We also study generalizations of
A-polynomials associated to $Q_{K}$ and (doubly) refined version of LMOV
invariants.
