  Under certain hypotheses, we show that Legendrian surfaces related by disk
surgery will have q-deformed augmentation spaces that are related by q-deformed
cluster transformation. The proof is geometric, via considerations of moduli of
holomorphic curves. In fact, our results naturally give a more general HOMFLYPT
"skein-valued cluster transformation", of which the q-cluster transformation is
the U(1) specialization.
  We apply our methods to the Legendrians associated to cubic planar graphs,
where mutation of graphs lifts to Legendrian disk surgery. We show that their
skein-valued mirrors transform by skein-valued cluster transformation, and give
a formula for the skein-valued curve counts on their fillings.
