The SL(3) web basis is a distinguished diagrammatic basis for certain spaces of tensor invariants developed by Kuperberg and Khovanov in the late 90's as a tool for computing quantum link invariants. Since then, it has found connections and applications to quantum topology, cluster algebras, dimer models, representation theory, and combinatorics. A main open problem has remained: how to find a basis replicating the desirable properties of this basis for SL(4) and beyond?

I will describe recent work using the combinatorics of modified versions of plabic graphs and the six-vertex model to unify all known constructions of such bases and to produce the first such basis for SL(4). I’ll end by highlighting the potential of this new framework to produce SL(n) bases in full generality.