Cube moves for $s$-embeddings and $\alpha$-realizations

Chelkak introduced $s$-embeddings as tilings by tangential quads which provide the right setting to study Ising model with arbitrary coupling constants on planar graphs. We prove existence and uniqueness of a local transformation for called cube move, consists in flipping three quadrilaterals such way that resulting tiling is also class $s$-embeddings. In passing, we give new simpler formula change star-triangle conjugated move introduce more generally $\alpha$-embeddings portion plane side lengths each quadrilateral $ABCD$ satisfy relation $AB^\alpha+CD^\alpha=AD^\alpha+BC^\alpha$, providing common generalization harmonic embeddings adapted resistor networks ($\alpha=2$) ($\alpha=1$). investigate properties these $\alpha$-embeddings.