Non-leaving-face property for marked surfaces

Abstract

We consider the polytope arising from a marked surface by flips of triangulations. Sleator, Tarjan and Thurston studied in 1988 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 establish that same non-leaving-face property for all unpunctured marked surfaces.arXiv:1801.09501v1 [math.CO]

On support $τ$-tilting graphs of gentle algebras

“…Clearly, the non-leaving-face property implies the reachable-in-face property. It is worth mentioning that for a 2-Calabi-Yau tilted gentle algebra A arising from a marked surface without punctures, Brüstle and Zhang [BZ2] have proved the non-leaving-face property for the supporting τ -tilting graph of A.…”

Exchange graphs of cluster algebras have the non‐leaving‐face property

“…The following definition is useful to investigate the non‐leaving‐face property (cf. [2, 6, 12, 13]). Definition Let $$\mathcal {G}$$ be an exchange graph and $$\mathcal {F}$$ a face of $$\mathcal {G}$$.…”

“…An abstract exchange graph has the structure of a generalized polytope and then the non‐leaving‐face property can be formulated in this general situation. In [2], the non‐leaving‐face property for exchange graphs arising from unpunctured marked surfaces was established. In [9], along with Zhou, we introduced another property called reachable‐in‐face property for abstract exchange graphs and proved that the exchange graph of a finite‐dimensional gentle algebra is connected and has the reachable‐in‐face property.…”