A Triangulation of the Flow Polytope of the Zigzag Graph

TL;DR

This paper explores the combinatorial structure of the flow polytope of the zigzag graph, revealing a grid subgraph structure and proposing new statistics that may determine its $h^*$-polynomial.

Contribution

It introduces a novel triangulation dual graph structure, links simplices to integer flows via a bijection, and develops new combinatorial statistics related to the flow polytope.

Findings

Dual graph of triangulation is a subgraph of a grid graph

Provides a numerical characterization of simplex adjacency

Proposes new statistics conjectured to recover the $h^*$-polynomial

Abstract

We show that the dual graph of the triangulation of the flow polytope of the zigzag graph adorned with the length-reverse-length framing is a subgraph of a grid graph. Through M\'esz\'aros, Morales, and Striker's bijection between simplices of the triangulation, integer flows of a different, supplemental flow polytope, we provide a simple numerical characterization of the adjacency between the triangulation's simplices in terms of their corresponding integer flows. The proofs result from the development of Postnikov and Stanley's sequences of noncrossing bipartite trees as combinatorial objects we call groves. We propose two new statistics derived from this construction that we conjecture recover the $h^*$-polynomial of the flow polytope of the zigzag graph.