标题： 可积动力学在射影几何中的 dimers 模型与圆柱上的三交叉图映射 显示英文标题

标题： Integrable dynamics in projective geometry via dimers and triple crossing diagram maps on the cylinder

摘要： 我们引入了扭转变三交叉图映射，这些是与圆柱面上的二分图相关联的射影空间中的点集，并利用它们为 Goncharov 和 Kenyon 从 toric dimer 模型构造的簇可积系统提供了几何实现。 利用这一概念，我们给出了五角星映射和交比动力学可积系统的几何证明，表明它们是簇可积系统。 我们证明，在适当的坐标下，交比动力学可以用几何$R$-矩阵来描述，从而解决了寻找描述交比动力学的簇代数结构这一公开问题。 显示英文摘要

摘要： We introduce twisted triple crossing diagram maps, collections of points in projective space associated to bipartite graphs on the cylinder, and use them to provide geometric realizations of the cluster integrable systems of Goncharov and Kenyon constructed from toric dimer models. Using this notion, we provide geometric proofs that the pentagram map and the cross-ratio dynamics integrable systems are cluster integrable systems. We show that in appropriate coordinates, cross-ratio dynamics is described by geometric $R$-matrices, which solves the open question of finding a cluster algebra structure describing cross-ratio dynamics.