Fan Realizations for Some 2-Associahedra

Abstract

ABSTRACTA 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.