Realizations of Multiassociahedra via Rigidity.
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| Title: | Realizations of Multiassociahedra via Rigidity. |
|---|---|
| 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] |
| Copyright of Discrete & Computational Geometry is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
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| Items | – Name: Title Label: Title Group: Ti Data: Realizations of Multiassociahedra via Rigidity. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Crespo+Ruiz%2C+Luis%22">Crespo Ruiz, Luis</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> luis.cresporuiz@unican.es</i><br /><searchLink fieldCode="AR" term="%22Santos%2C+Francisco%22">Santos, Francisco</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> francisco.santos@unican.es</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Discrete+%26+Computational+Geometry%22">Discrete & Computational Geometry</searchLink>. Jun2025, Vol. 73 Issue 4, p973-1015. 43p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Computational+geometry%22">Computational geometry</searchLink><br /><searchLink fieldCode="DE" term="%22Matroids%22">Matroids</searchLink><br /><searchLink fieldCode="DE" term="%22Logical+prediction%22">Logical prediction</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics%22">Mathematics</searchLink><br /><searchLink fieldCode="DE" term="%22Spheres%22">Spheres</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: 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] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Discrete & Computational Geometry is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1007/s00454-024-00698-y Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 43 StartPage: 973 Subjects: – SubjectFull: Computational geometry Type: general – SubjectFull: Matroids Type: general – SubjectFull: Logical prediction Type: general – SubjectFull: Mathematics Type: general – SubjectFull: Spheres Type: general Titles: – TitleFull: Realizations of Multiassociahedra via Rigidity. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Crespo Ruiz, Luis – PersonEntity: Name: NameFull: Santos, Francisco IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 06 Text: Jun2025 Type: published Y: 2025 Identifiers: – Type: issn-print Value: 01795376 Numbering: – Type: volume Value: 73 – Type: issue Value: 4 Titles: – TitleFull: Discrete & Computational Geometry Type: main |
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