  We establish a quadratic identity for the Yamada polynomial of ribbon cubic
graphs in 3-space, extending the Tutte golden identity for planar cubic graphs.
An application is given to the structure of the flow polynomial of cubic graphs
at zero. The golden identity for the flow polynomial is conjectured to
characterize planarity of cubic graphs, and we prove this conjecture for a
certain infinite family of non-planar graphs.
  Further, we establish exponential growth of the number of chromatic
polynomials of planar triangulations, answering a question of D. Treumann and
E. Zaslow. The structure underlying these results is the chromatic algebra, and
more generally the SO(3) topological quantum field theory.
