  It has been known since the work of Tutte that the value of the chromatic
polynomial of planar triangulations at $(3+\sqrt{5})/2$ has a number of
remarkable properties. We investigate to what extent Tutte's relations
characterize planar graphs. A version of the Tutte linear relation for the flow
polynomial at $(3-\sqrt{5})/2$ is shown to give a planarity criterion for
$3$-connected cubic graphs. A conjecture is formulated that the golden identity
for the flow polynomial characterizes planarity of cubic graphs as well. In
addition, Tutte's upper bound on the chromatic polynomial of planar
triangulations at $(3+\sqrt{5})/2$ is generalized to other Beraha numbers, and
an exponential lower bound is given for the value at $(3-\sqrt{5})/2$. The
proofs of these results rely on the structure of the Temperley-Lieb algebra and
more generally on methods of topological quantum field theory.
