  Using the flow property of the R-matrix defining the colored Jones
polynomial, we establish a natural bijection between the set of states on the
part arc-graph of a link projection and the set of states on a corresponding
bichromatic digraph, called arc-graph, as defined by Garoufalidis and Loebl. We
use this to give a new and essentially elementary proof for a knot state-sum
formula of Garoufalidis and Loebl. We will show that the state-sum
contributions of states on the part arc-graph defined by the universal R-matrix
of $U_q(sl(2,\mathbb{C}))$ correspond, under our bijection of sets of states,
to the contributions in the formula of Garoufalidis and Loebl. This will show
that the two state models are in fact not essentially distinct. Our approach
will also extend the formula of Garoufalidis and Loebl to links. This requires
some additional non-trivial observations concerning the geometry of states on
part arc-graphs. We will discuss in detail the computation of the arc-graph
state-sum, in particular for 3-braid closures.
