  Recently, a combinatorial interpretation of Baldoni and Vergne's generalized
Lidskii formula for the volume of a flow polytope was developed by Benedetti et
al.. This converts the problem of computing Kostant partition functions into a
problem of enumerating a set of objects called unified diagrams. We devise an
enhanced version of this combinatorial model to compute the volumes of flow
polytopes defined on a family of graphs called the k-caracol graphs, resulting
in the first application of the model to non-planar graphs. At k=1 and k=n-1,
we recover results for the classical caracol graph and the Pitman--Stanley
graph. Furthermore, we introduce the notion of in-degree gravity diagrams for
flow polytopes, which are equinumerous with (out-degree) gravity diagrams
considered by Benedetti et al.. We show that for the k-caracol flow polytopes,
these two kinds of gravity diagrams satisfy a natural combinatorial
correspondence, which raises an intriguing question on the relationship in the
geometry of two related polytopes.
