  We introduce graph potentials, which are Laurent polynomials associated to
(colored) trivalent graphs. We show that the birational type of the graph
potential only depends on the homotopy type of the colored graph, and use this
to define a topological quantum field theory. A similar construction was
recently introduced independently by Kontsevich--Odesskii under the name of
multiplicative kernels. We end our paper by giving an efficient computational
method to compute its partition function. This is the first paper in a series,
and we give a survey of the applications of graph potentials in the other
parts.
