We continue investigating the generalisations of geometrical statistical models introduced in [13], in the form of models of webs on the hexagonal lattice H having a U_q(sl_n) quantum group symmetry. We focus here on the n=3 case of cubic webs, based on the Kuperberg A_2 spider, and illustrate its properties by comparisons with the well-known dilute loop model (the n=2 case) throughout. A local vertex-model reformulation is exhibited, analogous to the correspondence between the loop model and a three-state vertex model. The n=3 representation uses seven states per link of H, displays explicitly the geometrical content of the webs and their U_q(sl_3) symmetry, and permits us to study the model on a cylinder via a local transfer matrix. A numerical study of the central charge reveals that for each q $in mathbb{C}$ in the critical regime, |q|=1, the web model possesses a dense and a dilute critical point, just like its loop model counterpart. In the dense $q=-e^{i pi/4}$ case, the n=3 webs can be identified with spin interfaces of the critical three-state Potts model defined on the triangular lattice dual to H. We also provide another mapping to a $mathbb{Z}_3$ spin model on H itself, using a high-temperature expansion. We then discuss the sector structure of the transfer matrix, for generic q, and its relation to defect configurations in both the strip and the cylinder geometries. These defects define the finite-size precursors of electromagnetic operators. This discussion paves the road for a Coulomb gas description of the conformal properties of defect webs, which will form the object of a subsequent paper. Finally, we identify the fractal dimension of critical webs in the $q=-e^{i pi/3}$ case, which is the n=3 analogue of the polymer limit in the loop model.
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