Mixed network calculus

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