Planar bicolored (plabic) graphs are combinatorial objects introduced by Postnikov to give parameterizations of the positroid cells of the totally nonnegative Grassmannian \(\text {Gr}^{\ge 0}(n,k)\). 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.

中文翻译：

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二次曲线图中的翻转周期

平面双色（素色）图是Postnikov引入的组合对象，用于对完全非负的Grassmannian \（\ text {Gr} ^ {\ ge 0}（n，k）\）的正片形单元进行参数化。可以通过一系列特定的动作来关联同一正片状细胞的任意两个幅图。所述倒装图表具有plabic图表为顶点，并具有连接其由单个移动相关plabic曲线边缘。Galashin的最新结果表明，斜方图可以看作是环状zonotope Z（n，3）。从这个角度来看，我们显示出翻转图的基本组是由长度为4、5和10的循环生成的，并使用此结果证明了Dylan Thurston关于三重交叉图的相关猜想。我们还将我们的结果应用到广义Baues问题的实例上。