  We continue our study of noncommutative resolutions of Coulomb branches in
the case of quiver gauge theories. These include the Slodowy slices in type A
and symmetric powers in $\mathbb{C}^2$ as special cases. These resolutions are
based on vortex line defects in quantum field theory, but have a precise
mathematical description, which in the quiver case is a modification of the
formalism of KLRW algebras. While best understood in a context which depends on
the geometry of the affine Grassmannian and representation theory in
characteristic $p$, we give a description of the Coulomb branches and their
commutative and non-commutative resolutions which can be understood purely in
terms of algebra.
  This allows us to construct a purely algebraic version of the knot homology
theory defined using string theory by Aganagi\'c, categorifying the
Reshetikhin-Turaev invariants for minuscule representations of type ADE Lie
algebras. We show that this homological invariant agrees with the
categorification of these invariants previously defined by the author, and thus
with Khovanov-Rozansky homology in type A.
