Reference : Once punctured disks, non-convex polygons, and pointihedra
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Physical, chemical, mathematical & earth Sciences : Mathematics
Once punctured disks, non-convex polygons, and pointihedra
Parlier, Hugo mailto [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit]
Pournin, Lionel [> >]
[en] Mathematics - Combinatorics ; Mathematics - Geometric Topology
[en] We explore several families of flip-graphs, all related to polygons or punctured polygons. In particular, we consider the topological flip-graphs of once-punctured polygons which, in turn, contain all possible geometric flip-graphs of polygons with a marked point as embedded sub-graphs. Our main focus is on the geometric properties of these graphs and how they relate to one another. In particular, we show that the embeddings between them are strongly convex (or, said otherwise, totally geodesic). We also find bounds on the diameters of these graphs, sometimes using the strongly convex embeddings. Finally, we show how these graphs relate to different polytopes, namely type D associahedra and a family of secondary polytopes which we call pointihedra.
24 pages, 6 figures

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