In the 1960s Tutte observed that the value of the chromatic polynomial of planar triangulations at (golden ratio +1) obeys a number of remarkable properties. In this talk I will explain how TQFT gives rise to a conceptual framework for studying planar triangulations. I will discuss several extensions of Tutte’s results and applications to the structure of the chromatic and flow polynomials of graphs, and the Yamada polynomial of spatial graphs. This talk is based on joint works with Ian Agol and with Paul Fendley.