Kuperberg’s SL_3 non-elliptic web basis consists of certain trivalent planar graphs. Fontaine--Kamnitzer--Kuperberg showed that their duals may be realized as subcomplexes of the affine building Delta(PGL_3). The result is a collection of CAT(0) triangulated surfaces related to the geometric Satake correspondence.

Recently, an SL(4) web basis was introduced by Gaetz--Pechenik--Pfannerer--Striker--S. which comes with "moves". We show the moves may be understood geometrically as forming "pockets", certain highly structured 3D simplicial subcomplexes of Delta(PGL_4). Special cases correspond to plane partitions, alternating sign matrices, tilings of the Aztec diamond, and more. Joint with Christian Gaetz, Jessica Striker, and Haihan Wu.