  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.
