On the Common Generalization of Gentle Algebras and Framed Directed Acyclic Graphs

In the study of flow polytopes, a directed acyclic graph (DAG) with a choice of framing gives a regular unimodular triangulation on its space of unit nonnegative flows. In representation theory, a gentle algebra has recently been equipped with a space of unit flows admitting triangulation and subdivision results capturing its tau-tilting theory. These theories from different areas of mathematics overlap: flows on gently framed DAGs are the same as flows on paired representation-finite gentle algebras. In this article we develop the common generalization of these two theories by defining (framed) turbulence charts, which may be thought of as analogs of (framed) DAGs without the conditions of (D)irectedness and (A)cyclicity. The space of unit flows on a turbulence chart is its turbulence polyhedron. We give presentation, subdivision, and triangulation results on turbulence polyhedra which restrict to known results in the settings of framed DAGs and gentle algebras.