  We show how to combine higgsed topological vertices introduced in our
previous work with conventional refined topological vertices. We demonstrate
that the extended formalism describes very general interacting D5-NS5-D3 brane
systems. In particular, we introduce new types of intertwining operators of
Ding-Iohara-Miki algebra between different types of Fock representations
corresponding to the crossings of NS5 and D5 branes. As a byproduct we obtain
an algebraic description of the Hanany-Witten brane creation effect, give an
efficient recipe to compute the brane factors in 3d N=2 and N=4 quiver gauge
theories and demonstrate how 3d S-duality appears in our setup.
