平面图中的翻转圈
Flip cycles in plabic graphs
摘要：
平面双色图是由Postnikov引入的一个组合对象，它给出了完全非负Grassmannian $\text{Gr}^{\geq 0}(n,k)$的正素胞的参数化。同一个正向胞腔的任意两个平面图可以通过一系列特定的移动而联系起来。翻转图的顶点是plabic图，并且翻转图的边连接通过单个移动而相关的plabic图。Galashin最近的一个结果表明，对于循环带形$Z(n,3)$，可以把平面图看作是带形镶嵌的横截面。从这个角度来看，我们证明了翻转图的基本群是由长度为4，5和10的圈生成的，并利用这个结果证明了Dylan Thurston关于三重交叉图的一个相关猜想。我们也应用我们的结果取得进展的一个例子的广义Baues问题。
英文摘要：
Planar bicolored (plabic) graphs are combinatorial objects introduced by Postnikov to give parameterizations of the positroid cells of the totally nonnegative Grassmannian $\text{Gr}^{\geq 0}(n,k)$. Any two plabic graphs for the same positroid cell can be related by a sequence of certain moves. The flip graph has plabic graphs as vertices and has edges connecting the plabic graphs which are related by a single move. A recent result of Galashin shows that plabic graphs can be seen as cross-sections of zonotopal tilings for the cyclic zonotope $Z(n,3)$. Taking this perspective, we show that the fundamental group of the flip graph is generated by cycles of length 4, 5, and 10, and use this result to prove a related conjecture of Dylan Thurston about triple crossing diagrams. We also apply our result to make progress on an instance of the generalized Baues problem.
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