Realizations of Multiassociahedra via Rigidity.
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| Title: | Realizations of Multiassociahedra via Rigidity. |
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| Authors: | Crespo Ruiz, Luis1 (AUTHOR) luis.cresporuiz@unican.es, Santos, Francisco1 (AUTHOR) francisco.santos@unican.es |
| Source: | Discrete & Computational Geometry. Jun2025, Vol. 73 Issue 4, p973-1015. 43p. |
| Subjects: | Computational geometry, Matroids, Logical prediction, Mathematics, Spheres |
| Abstract: | Let Δ k (n) denote the simplicial complex of (k + 1) -crossing-free subsets of edges in [ n ] 2 . Here k , n ∈ N and n ≥ 2 k + 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 - 2 k - 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 ≤ 2 k + 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 Δ k (n) as a polytope for (k , n) ∈ { (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 ≥ 2 k + 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. [ABSTRACT FROM AUTHOR] |
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| Database: | Engineering Source |
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| Abstract: | Let Δ k (n) denote the simplicial complex of (k + 1) -crossing-free subsets of edges in [ n ] 2 . Here k , n ∈ N and n ≥ 2 k + 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 - 2 k - 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 ≤ 2 k + 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 Δ k (n) as a polytope for (k , n) ∈ { (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 ≥ 2 k + 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. [ABSTRACT FROM AUTHOR] |
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| ISSN: | 01795376 |
| DOI: | 10.1007/s00454-024-00698-y |
