Fan Realizations for Some 2-Associahedra

ABSTRACT

A k-associahedron is a simplicial complex whose facets, called k-triangulations, are the inclusion maximal sets of diagonals of a convex polygon where no k + 1 diagonals mutually cross. Such complexes are conjectured for about a decade to have realizations as convex polytopes, and therefore as complete simplicial fans. Apart from four one-parameter families including simplices, cyclic polytopes, and classical associahedra, only two instances of multiassociahedra have been geometrically realized so far. This article reports on conjectural realizations for all 2-associahedra, obtained by heuristic methods arising from natural geometric intuition on subword complexes. Experiments certify that we obtain fan realizations of 2-associahedra of an n-gon for n ∈ {10, 11, 12, 13}, further ones being out of our computational reach.

KEYWORDS:

• fans
• multiassociahedra
• multitriangulations
• subword complexes

2000 AMS SUBJECT CLASSIFICATION:

• 52B11
• 52B12
• 52B40
• 05E45

Notes

1 http://www.lix.polytechnique.fr/∼manneville/homePageEnglishResearch.html

2 This is in fact a definition of spherical subword complexes of type A_n.

3 Since $\mathcal{S}\left(\mathsf{Q}\right)$ and $\mathcal{S}\left({\mathsf{Q}}^{-1}\right)$ are isomorphic.