Equivalence of the Potts model or Whitney polynomial with an ice-type model

The partition function of the Potts model (1952) on any lattice can readily be written as a Whitney polynomial (1932). Temperley and Lieb (Proc. R. Soc., vol.A322, p.251 of 1971) have used operator methods to show that, for a square lattice, this problem is in turn equivalent to a staggered ice-type model. Here the authors rederive this equivalence by a graphical method, which they believe to be simpler, and which applies to any planar lattice. For instance, they also show that the Potts model on the triangular or honeycomb lattice is equivalent to an ice-type model on a Kagome lattice.