  We consider a self-avoiding walk model (SAW) on the faces of the square
lattice $\mathbb{Z}^2$. This walk can traverse the same face twice, but crosses
any edge at most once. The weight of a walk is a product of local weights: each
square visited by the walk yields a weight that depends on the way the walk
passes through it. The local weights are parametrised by angles
$\theta\in[\frac{\pi}{3},\frac{2\pi}{3}]$ and satisfy the Yang-Baxter equation.
The self-avoiding walk is embedded in the plane by replacing the square faces
of the grid with rhombi with corresponding angles.
  By means of the Yang-Baxter transformation, we show that the 2-point function
of the walk in the half-plane does not depend on the rhombic tiling (i.e. on
the angles chosen). In particular, this statistic coincides with that of the
self-avoiding walk on the hexagonal lattice. Indeed, the latter can be obtained
by choosing all angles $\theta$ equal to $\frac{\pi}{3}$.
  For the hexagonal lattice, the critical fugacity of SAW was recently proved
to be equal to $1+\sqrt{2}$. We show that the same is true for any choice of
angles. In doing so, we also give a new short proof to the fact that the
partition function of self-avoiding bridges in a strip of the hexagonal lattice
tends to 0 as the width of the strip tends to infinity. This proof also yields
a quantitative bound on the convergence.
