Non-leaving-face property for marked surfaces
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We consider the polytope arising from a marked surface by flips of triangulations. D. D. Sleator, R. E. Tarjan, and W. P. Thurston [J. Amer. Math. Soc., 1988, 1(3): 647–681] studied the diameter of the associahedron, which is the polytope arising from a marked disc by flips of triangulations. They showed that every shortest path between two vertices in a face does not leave that face. We give a new method, which is different from the one used by V. Disarlo and H. Parlier [arXiv: 1411.4285] to establish the same non-leaving-face property for all unpunctured marked surfaces.
KeywordsMarked surface non-leaving-face property exchange graph
MSC13F60 13E10 16G20
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The authors would like to thank Lionel Pournin for pointing out the work of Disarlo and Parlier  to them. This work was supported in part by the National Natural Science Foundation of China (Grant No. 11401022).
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