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Fan realizations for some 2-associahedra. (English) Zbl 1406.52026
Summary: 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 \(n\in \{10, 11, 12, 13\}\), further ones being out of our computational reach.

MSC:
52B11 \(n\)-dimensional polytopes
52B12 Special polytopes (linear programming, centrally symmetric, etc.)
52B40 Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.)
05E45 Combinatorial aspects of simplicial complexes
Software:
SageMath
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