Recent advances in Quantum Topology assign \(q\)-series to knots in at least three different ways. The \(q\)-series are given by generalized Nahm sums (i.e., special \(q\)-hypergeometric sums) and have unknown modular and asymptotic properties. We give an efficient method to compute those \(q\)-series that come from planar graphs (i.e., reduced Tait graphs of alternating links) and compute several terms of those series for all graphs with at most 8 edges drawing several conclusions. In addition, we give a graph-theory proof of a theorem of Dasbach-Lin which identifies the coefficient of \(q^k\) in those series for \(k=0,1,2\) in terms of polynomials on the number of vertices, edges and triangles of the graph. Updated tables of data.