Abstract
We give a new description of the combinatorics of triangulations of even-dimensional cyclic polytopes, and of their bistellar flips. We show that the tropical exchange relation governing the number of intersections between diagonals of a polygon and a lamination (which generalizes to arbitrary surfaces) can also be generalized in a different way, to the setting of higher dimensional cyclic polytopes. Nous donnons une nouvelle description de la combinatoire des triangulations des polytopes cycliques, et de leurs mouvements bistellaires. Nous démontrons que la relation d’échange qui gouverne le nombre d'intersections entre les diagonaux d'une polygone et une lamination (qui peut être généralisée à une surface arbitraire) peut également être généralisée au cadre des polytopes cycliques.
Highlights
By a triangulation of a m-gon, we mean a subdivision of the m-gon into triangles, by adding diagonals which do not cross in the interior of the m-gon
There is a natural graph structure on the set of triangulations of an m-gon: two triangulations are adjacent iff they are related by a single diagonal flip in which a diagonal is removed, and replaced by the other diagonal of the quadrilateral which results
We say that two triangulations S and T of P are related by a bistellar flip if there are some e + 2 points of P such that S and T coincide outside the convex hull of these points, and restricted to the convex hull of these points, S and T agree with the two different triangulations of these points
Summary
By a triangulation of a (convex) m-gon, we mean a subdivision of the m-gon into triangles, by adding diagonals which do not cross in the interior of the m-gon. There is one generalization which is well-known, in which we remove the convexity, and replace diagonals by isotopy classes of curves in an arbitrary surface, with marked points on its boundary components where the curves are required to begin and end, and consider the set of triangulations that arises in this way. This approach is used to define the cluster algebra associated to such a surface (provided the surface is orientable). Complete proofs, the reader is referred to that paper
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