We associate to each ciliated bipartite ribbon graph in R^3 an isotopy invariant Laurent polynomial in a single variable q^(1/N), called the SL_N quantum trace, which can be expressed as a quantum deformation of the partition function for the N-dimer model. The construction is based on Sikora’s SL_N quantum traces for N-webs in R^3. For planar graphs, the quantum trace is moreover a symmetric Laurent polynomial in q, which can be expressed as a quantum deformation of the Kasteleyn determinant of the graph equipped with the trivial connection. We also provide a similar expression for planar graphs equipped with a general quantum matrix connection (subject to a relatively strong commutativity constraint). This is joint work with Richard Kenyon, Nicholas Ovenhouse, Sam Panitch, and Sri Tata.