Non-leaving-face property for marked surfaces

  • Thomas Brüstle
  • Jie ZhangEmail author
Research Article


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.


Marked surface non-leaving-face property exchange graph 


13F60 13E10 16G20 


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The authors would like to thank Lionel Pournin for pointing out the work of Disarlo and Parlier [8] to them. This work was supported in part by the National Natural Science Foundation of China (Grant No. 11401022).


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Copyright information

© Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Département de MathématiquesUniversité de SherbrookeSherbrookeCanada
  2. 2.School of Mathematics and StatisticsBeijing Institute of TechnologyBeijingChina

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