Realizations of multiassociahedra via rigidity

TL;DR

This paper explores the geometric realizations of multiassociahedra complexes using rigidity theory, providing new polytopal and fan realizations for specific parameters and establishing limitations for such realizations.

Contribution

It introduces novel realizations of $ ext{multiassociahedra}$ as polytopes and fans for certain parameters, and proves non-realizability results for larger cases using rigidity theory.

Findings

Realizes $ ext{multiassociahedra}$ as polytopes for specific $(k,n)$ values.

Provides simplicial fan realizations for all $n extless=13$ with some exceptions.

Shows non-realizability for larger $(k,n)$ using rigidity theory constraints.

Abstract

Let $\Delta_k(n)$ denote the simplicial complex of $(k+1)$-crossing-free subsets of edges in $\binom{[n]}{2}$. Here $k,n\in \mathbb N$ and $n\ge 2k+1$. Jonsson (2003) proved that (neglecting the short edges that cannot be part of any $(k+1)$-crossing), $\Delta_k(n)$ is a shellable sphere of dimension $k(n-2k-1)-1$, and conjectured it to be polytopal. The same result and question arose in the work of Knutson and Miller (2004) on subword complexes. Despite considerable effort, the only values of $(k,n)$ for which the conjecture is known to hold are $n\le 2k+3$ (Pilaud and Santos, 2012) and $(2,8)$ (Bokowski and Pilaud, 2009). Using ideas from rigidity theory and choosing points along the moment curve we realize $\Delta_k(n)$ as a polytope for $(k,n)\in \{(2,9), (2,10) , (3,10)\}$. We also realize it as a simplicial fan for all $n\le 13$ and arbitrary $k$, except the pairs $(3,12)$ and $(3,13)$. Finally, we also show that for $k\ge 3$ and $n\ge 2k+6$ no choice of points can realize $\Delta_k(n)$ via bar-and-joint rigidity with points along the moment curve or, more generally, via cofactor rigidity with arbitrary points in convex position.