Feynman diagrams were introduced by physicists. They arise naturally in mathematics (from knots and singular knots), and in molecular biology (from RNA folding). In particular, work of G. Vernizzi, H. Orland, and A. Zee has shown that the "genus" of Feynman diagrams plays an important role in the prediction of RNA structures. The transition polynomial for 4-regular graphs was defined by Jaeger to unify polynomials given by vertex reconfigurations similar to the skein relations of knots. It is closely related to the Kauffman bracket, Tutte polynomial, and the Penrose polynomial. We define a transition polynomial for Feynman diagrams and discuss its properties. In particular, we show that the genus of a Feynman diagram is encoded in the transition polynomial. This is joint work with Kerry Luse.