Framing Triangulations and Framing Posets of Planar DAGs with Nontrivial Netflow Vectors

TL;DR

This paper extends the theory of flow polytopes and their triangulations from simple DAGs with one source and sink to more complex DAGs with multiple sources, sinks, and nontrivial netflow vectors, under planarity conditions.

Contribution

It introduces a new unimodular triangulation for flow polytopes of complex DAGs and defines a poset structure on the simplices, generalizing previous models.

Findings

Constructed a unimodular triangulation indexed by combinatorial data.

Established a poset structure on the maximal simplices.

Generalized the Tamari lattice and weak order to more complex DAGs.

Abstract

The space of unit flows on a directed acyclic graph (DAG) with one source and one sink is known to admit regular unimodular triangulations induced by framings of the DAG. The dual graph of any of these triangulations may be given the structure of the Hasse diagram of a lattice, generalizing many variations of the Tamari lattice and the weak order. We extend this theory to flow polytopes of DAGs which may have multiple sources, multiple sinks, and nontrivial netflow vectors under certain planarity conditions. We construct a unimodular triangulation of such a flow polytope indexed by combinatorial data and give a poset structure on its maximal simplices generalizing the strongly planar one-source-one-sink case.