Realizations of multiassociahedra via bipartite rigidity

TL;DR

This paper investigates the polytopality of multiassociahedra complexes using bipartite rigidity matrices, providing realizations for certain cases and proving non-realizability in others, with algebraic and tropical geometric insights.

Contribution

It introduces a novel approach using bipartite rigidity matrices to realize multiassociahedra as polytopes or fans, advancing understanding of their geometric structure.

Findings

Realizes $Ass_k(n)$ as a polytope for $k=2$, $n extless=10$

Realizes $Ass_k(n)$ as a fan for $k=2$, $n extless=13$ and $k=3$, $n extless=11$

Proves non-realizability for $k extgreater=3$, $n extgreater=12$ or $2k+4$

Abstract

Let $Ass_k(n)$ denote the simplicial complex of $(k+1)$-crossing-free subsets of edges in $\binom{n}{2}$. Here $k,n\in \mathbb{N}$ and $n\ge 2k+1$. It is conjectured that this simplicial complex is polytopal (Jonsson 2005). However, despite several recent advances, this is still an open problem. In this paper we attack this problem using as a vector configuration the rows of a rigidity matrix, namely, hyperconnectivity restricted to bipartite graphs. We see that in this way $Ass_k(n)$ can be realized as a polytope for $k=2$ and $n\le 10$, and as a fan for $k=2$ and $n\le 13$, and for $k=3$ and $n\le 11$. However, we also prove that the cases with $k\ge 3$ and $n\ge \max\{12,2k+4\}$ are not realizable in this way. We also give an algebraic interpretation of the rigidity matroid, relating it to a projection of determinantal varieties with implications in matrix completion, and prove the presence of a fan isomorphic to $Ass_{k-1}(n-2)$ in the tropicalization of that variety.