We introduce a family of geometrical lattice models generalising the well-known loop model on the hexagonal lattice. These models have a $U_q(sl_n)$ quantum group symmetry, the loop model being the $n=2$ case. The general models give rise to branching webs and describe, at a special point, the interfaces in $Z_n$ symmetric spin models. We mainly discuss the $n=3$ case of bipartite cubic webs, which is based on the Kuperberg $A_2$ spider. We exhibit a local vertex-model reformulation, analogous to the well-known correspondence between the loop model and the nineteen-vertex model. The local formulation allows us in particular to study the model by means of transfer matrices and conformal field theory. We find that it has a rich phase diagram, including a dense and a dilute phase that generalise those known for the loop model. Based on joint work with Augustin Lafay and Azat Gainutdinov (arXiv:2101.00282 and 2107.10106).[-]