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arXiv · 1703.09953

Geometric realizations of the accordion complex of a dissection

Thibault Manneville·Vincent Pilaud
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Abstract

Consider $2n$ points on the unit circle and a reference dissection $\mathrm{D}_\circ$ of the convex hull of the odd points. The accordion complex of $\mathrm{D}_\circ$ is the simplicial complex of non-crossing subsets of the diagonals with even endpoints that cross a connected subset of diagonals of $\mathrm{D}_\circ$. In particular, this complex is an associahedron when $\mathrm{D}_\circ$ is a triangulation and a Stokes complex when $\mathrm{D}_\circ$ is a quadrangulation. In this paper, we provide geometric realizations (by polytopes and fans) of the accordion complex of any reference dissection $\mathrm{D}_\circ$, generalizing known constructions arising from cluster algebras.

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Thibault Manneville, Vincent Pilaud. 2017-03-29. Geometric realizations of the accordion complex of a dissection. https://doi.org/10.1007/s00454-018-0004-2

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