This talk will outline a circle of ideas at the intersection of quantum topology, combinatorics, and lattice models in statistical mechanics. I will explain how the structure of (2+1)-dimensional topological quantum field theory gives rise to a conceptual framework for studying planar triangulations. More generally, applications will be given to the structure of classical and quantum polynomial invariants of graphs on surfaces and in 3-space. (No prior knowledge of quantum topology will be assumed.) This talk is based on joint works with Paul Fendley and with Ian Agol.