  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.
