Low dimensional flow polytopes and their toric ideals

TL;DR

This paper investigates the algebraic and combinatorial properties of flow polytopes, revealing that their toric ideals have initial ideals generated by low-degree square-free monomials, with specific exceptions in four dimensions.

Contribution

It classifies compressed flow polytopes up to dimension four, computes their Ehrhart polynomials, and introduces a method to analyze their toric ideals and triangulations.

Findings

Toric ideal initial ideals are generated by square-free monomials of degree at most d.

Flow polytopes of dimension ≤4 have initial ideals generated by degree ≤2, except the 4D Birkhoff polytope.

Classification and Ehrhart polynomial computation for compressed flow polytopes up to dimension four.

Abstract

The toric ideal of a $d$-dimensional flow polytope has an initial ideal generated by square-free monomials of degree at most $d$. The toric ideal of a flow polytope of dimension at most four has an initial ideal generated by square-free monomials of degree at most two, with the only exception of the four-dimensional Birkhoff polytope, whose toric ideal has an initial ideal generated by a square-free cubic monomial. The proof is based on a method to classify certain compressed flow polytopes, and a construction of a quadratic pulling triangulation of them. Along the way compressed flow polytopes are classified up to dimension four, and their Ehrhart polynomials are computed.