The complex of non-crossing diagonals of a polygon

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Given a convex n-gon P in R{double-struck}2 with vertices in general position, it is well known that the simplicial complex θ(P) with vertex set given by diagonals in P and facets given by triangulations of P is the boundary complex of a polytope of dimension n-3. We prove that for any non-convex polygonal region P with n vertices and h+1 boundary components, θ(P) is a ball of dimension n+3h-4. We also provide a new proof that θ(P) is a sphere when P is convex with vertices in general position.

Original languageEnglish
Pages (from-to)642-649
Number of pages8
JournalJournal of Combinatorial Theory. Series A
Issue number6
StatePublished - Aug 2010

Bibliographical note

Funding Information:
E-mail addresses: (B. Braun), (R. Ehrenborg). URLs: (B. Braun), (R. Ehrenborg). 1 Partially supported by National Science Foundation grant DMS-0758321. 2 Partially supported by National Security Agency grant H98230-06-1-0072.


  • Associahedra
  • Discrete Morse theory
  • Non-convex polygon
  • Simplicial complex

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics


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