Flows on cyclic graphs, complete gentle algebras, and the doppelgänghedron

Flow cones of a directed acyclic graph admit a family of unimodular triangulations given by Danilov, Karzanov, and Koshevoy (DKK) whose normal fans are related to (generalizations) of the associahedron and permutahedron. A correspondence between these triangulations for certain graphs and maximal cones of a g-vector fan of a gentle quiver associated to the graph was discovered by von Bell, Braun, Bruegge, Hanely, Peterson, Sherhiyenko and Yip in 2022. This correspondence has been fruitful to uncover lattice structures in the triangulations. We consider flow cones of certain graphs with cycles. For this case, we give a DKK-like triangulation of the cone, and extend their correspondence to the finite g-vector fan of locally gentle quivers of the graphs. In addition, we extend to cyclic graphs a mysterious result of Postnikov--Stanley and Baldoni--Vergne giving the volume of flow polytopes of acyclic graphs as the number of certain integer flows on the same graph. We illustrate our results with a two-parameter family of cyclic graphs which include a cycle graph and nested 2-cycles as special cases whose normal fans are related to the cyclohedron and a new polytope, called the doppelgänghedron, whose f-vector is equal to that of the permutahedron but which has a different combinatorial type.