Root polytopes, Tutte polynomials, and a duality theorem for bipartite graphs

TL;DR

This paper establishes a deep connection between the interior polynomial of a bipartite hypergraph, the Ehrhart polynomial of its root polytope, and the triangulation's h-vector, revealing a duality and recovering known identities.

Contribution

It proves the equivalence of the interior polynomial, Ehrhart polynomial, and h-vector for bipartite graphs, and demonstrates a duality between a graph and its transpose.

Findings

Interior polynomial equals Ehrhart polynomial of the root polytope

Interior polynomials of a bipartite graph and its transpose are equal

Special cases recover known hypergeometric identities and relate to Floer homology

Abstract

Let G be a connected bipartite graph with color classes E and V and root polytope Q. Regarding the hypergraph (V,E) induced by G, we prove that its interior polynomial is equivalent to the Ehrhart polynomial of Q, which in turn is equivalent to the h-vector of any triangulation of Q. It follows that the interior polynomials of (V,E) and its transpose (E,V) agree. When G is a complete bipartite graph, our result recovers a well known hypergeometric identity due to Saalsch\"utz. It also implies that certain extremal coefficients in the Homfly polynomial of a special alternating link can be read off of an associated Floer homology group.