Higher Secondary Polytopes for Two-Dimensional Zonotopes

TL;DR

This paper explores higher secondary polytopes for two-dimensional zonotopes, relating their structure to flip graphs of hypertriangulations and computing their diameters, thus advancing understanding of their combinatorial properties.

Contribution

It establishes a connection between the 1-skeletons of higher secondary polytopes and flip graphs of hypertriangulations in 2D, and calculates their diameters.

Findings

Relation between Minkowski sum 1-skeletons and hypertriangulation flip graphs

Explicit diameter calculations for these polytopes

Extension of higher secondary polytope theory to 2D zonotopes

Abstract

Very recently, Galashin, Postnikov, and Williams introduced the notion of higher secondary polytopes, generalizing the secondary polytope of Gelfand, Kapranov, and Zelevinsky. Given an $n$-point configuration $\mathcal{A}$ in $\mathbb{R}^{d-1}$, they define a family of convex $(n-d)$-dimensional polytopes $\widehat{\Sigma}_{1}, \ldots, \widehat{\Sigma}_{n-d}$. The $1$-skeletons of this family of polytopes are the flip graphs of certain combinatorial configurations which generalize triangulations of $\text{conv} \mathcal{A}$. We restrict our attention to $d=2$. First, we relate the $1$-skeleton of the Minkowski sum $\widehat{\Sigma}_{k} + \widehat{\Sigma}_{k-1}$ to the flip graph of "hypertriangulations" of the deleted $k$-sum of $\mathcal{A}$ when $\mathcal{A}$ consists of distinct points. Second, we compute the diameter of $\widehat{\Sigma}_{k}$ and $\widehat{\Sigma}_{k}+\widehat{\Sigma}_{k-1}$ for all $k$.