Signotopes Induce Unique Sink Orientations on Grids

TL;DR

This paper explores how Signotopes induce unique sink orientations on higher-dimensional grid products, extending previous characterizations from 2D to multi-dimensional cases and generalizing to signotopes.

Contribution

It generalizes the characterization of USOs induced by linear functions from 2D grids to higher dimensions and introduces the concept of Signotopes in this context.

Findings

Extended USO characterizations to products of multiple simplices.

Connected Signotopes with unique sink orientations in higher dimensions.

Provided a framework for understanding USOs on r-dimensional grids.

Abstract

A unique sink orientation (USO) is an orientation of the edges of a polytope in which every face contains a unique sink. For a product of simplices $\Delta_{m-1} \times \Delta_{n-1}$, Felsner, G\"artner and Tschirschnitz (2005) characterize USOs which are induced by linear functions as the USOs on a $(m \times n)$-grid that correspond to a two-colored arrangement of lines. We generalize some of their results to products $\Delta^1 \times\cdots\times \Delta^r$ of $r$ simplices, USOs on $r$-dimensional grids and $(r+1)$-signotopes.