Ashkin-Teller model on the iso-radial graphs

We find the critical surface of the Ashkin-Teller model on the generic iso-radial graphs by using the results for the anisotropic Ashkin-Teller model on the square lattice. Different geometrical aspects of this critical surface are discussed, especially their connection to the anisotropy angle. The free energy of the model on the generic iso-radial graph is extracted using the inversion identities. In addition, lattice holomorphic variables are discussed at some particular points of the critical line. We check our conjectures numerically for the anisotropic triangular lattice Ashkin-Teller model.