Signotopes Induce Unique Sink Orientations on 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 $Δ_{m-1} \times Δ_{n-1}$, Felsner, Gärtner 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 $Δ^1 \times\cdots\times Δ^r$ of $r$ simplices, USOs on $r$-dimensional grids and $(r+1)$-signotopes.

Figures (10)

• Figure 1: As a Cartesian product of simplices, the $3$-polytope $\Delta_1\times\Delta_2$ has 5 facets.
• Figure 2: Double twist $\text{DT}$; acyclic but non-admissible USO
• Figure 3: (a): Example of an admissible USO. Transitive arrows are not drawn. (b): Refined index of the orientation in (a).
• Figure 4: Each triple of pseudolines $i<j<k$ is either negative or positive.
• Figure 5: (a): Block colored pseudoline arrangement $\mathcal{A}$ of $r=5$ red and $b=5$ blue pseudolines. (b): Grid drawing of $\mathcal{A}$: Every crossing between red and blue pseudolines lies on a point of a $(5\times 5)$-grid.

Theorems & Definitions (19)

• Lemma 1: Theorem 2 in gaertnerMorrisRuest08
• Theorem 1: Felsner, Gärtner & Tschirschnitz felsnerGaertnerTschirschnitz05
• Theorem 2
• proof
• Theorem 3
• Lemma 2
• Lemma 3
• Lemma 4
• proof
• proof : Proof of Lemma \ref{['lemma:unique_sink_dim2']}