Realizations of multiassociahedra via rigidity

TL;DR

The paper advances the realization theory of multiassociahedra $\overline{\Delta}_{k}(n)$ by tying polytopality and complete-fan realizability to rigidity matroids. Using points on the moment curve, it achieves new polytopal realizations for $(2,9)$, $(2,10)$, and $(3,10)$ and broad fan realizations for $n\le 13$ (with two exceptions), while proving obstructions for $k\ge 3$ when $n\ge 2k+6$ that rules out moment-curve/cofactor realizations in those cases. The work develops both obstructions (via Morgan–Scott-type arguments in cofactor rigidity) and positive results (notably for $k=2$) by exploiting monotonicity, stars, flips, and regular liftings, with extensive computational experiments guiding the constructions. Overall, it strengthens the bridge between combinatorial topology of $k$-triangulations and geometric realizations via rigidity, and it delineates the boundary between polytopality and non-polytopality in this family. The findings also point toward a broader conjecture: realizations as a basis collection along the moment curve may hold generically for all $k,n$, which would imply wider polytopality results beyond the current proven cases.

Abstract

Let $Δ_k(n)$ 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), $Δ_k(n)$ 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 $Δ_k(n)$ 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 $Δ_k(n)$ via bar-and-joint rigidity with points along the moment curve or, more generally, via cofactor rigidity with arbitrary points in convex position.

Figures (2)

• Figure 1: Two configurations of six points in convex position, chosen along the parabola. The configuration in the left is positively oriented, the one on the right is in Desargues position
• Figure 2: Explanation of the last equivalence in the proof of Theorem \ref{['thm:cofact']}. Each of the inequalities $\alpha<\beta$ and $\alpha'>\beta'$ is equivalent to the configuration being positively oriented.

Theorems & Definitions (92)

• Conjecture 1.1: Jonsson
• Remark 1.2
• Theorem 1.3
• Conjecture 1.4
• Theorem 1.5
• Conjecture 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Remark 1.10