Abstract: Let dk(n) denote the simplicial complex of (k+1)-crossing-free subsets of edges in [n]2. Here k,nEN and n >=2k+1. Jonsson (2003) proved that [neglecting the short edges that cannot be part of any (k+1)-crossing], dk(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 (Adv Math 184(1):161-176, 2004) on subword complexes. Despite considerable effort, the only values of (k, n) for which the conjecture is known to hold are n<=2k+3 (Pilaud and Santos, Eur J Comb. 33(4):632?662, 2012. https://doi.org/10.1016/j.ejc.2011.12.003) and (2, 8) (Bokowski and Pilaud, On symmetric realizations of the simplicial complex of 3-crossing-free sets of diagonals of the octagon. In: Proceedings of the 21st annual Canadian conference on computational geometry, 2009). Using ideas from rigidity theory and choosing points along the moment curve we realize dk(n) as a polytope for (k,n)E{(2,9),(2,10),(3,10)}. We also realize it as a simplicial fan for all n<=13 and arbitrary k, except the pairs (3, 12) and (3, 13). Finally, we also show that for k>=3 and n>=2k+6 no choice of points can realize dk(n) via bar-and-joint rigidity with points along the moment curve or, more generally, via cofactor rigidity with arbitrary points in convex position.
