Realizations of multiassociahedra via rigidity

Let ${\mathrm{\Delta }}_k(n)$ denote the simplicial complex of $(k+1)$-crossing-free subsets of edges in $\left(\genfrac{}{}{0px}{}{[n]}{2}\right)$. 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), ${\mathrm{\Delta }}_k(n)$ 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 ${\mathrm{\Delta }}_k(n)$ 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 ${\mathrm{\Delta }}_k(n)$ via bar-and-joint rigidity with points along the moment curve or, more generally, via cofactor rigidity with arbitrary points in convex position.