  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.
