We classify weakly exact, rational Lagrangian tori in $T^* \mathbb{T}^2- 0_{\mathbb{T}^2}$ up to Hamiltonian isotopy. This result is related to the classification theory of closed $1$-forms on $\mathbb{T}^n$ and also has applications to symplectic topology. As a 1st corollary, we strengthen a result due independently to Eliashberg–Polterovich and to Giroux describing Lagrangian tori in $T^* \mathbb{T}^2-0_{\mathbb{T}^2}$, which are homologous to the zero section. As a 2nd corollary, we exhibit pairs of disjoint totally real tori $K_1, K_2 \subset T^*\mathbb{T}^2$, each of which is isotopic through totally real tori to the zero section, but such that the union $K_1 \cup K_2$ is not even smoothly isotopic to a Lagrangian. In the 2nd part of the paper, we study linking of Lagrangian tori in $({\mathbb{R}}^4, \omega )$ and in rational symplectic $4$-manifolds. We prove that the linking properties of such tori are determined by purely algebro-topological data, which can often be deduced from enumerative disk counts in the monotone case. We also use this result to describe certain Lagrangian embedding obstructions.

中文翻译：

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在辛 4 流形中连接拉格朗日环和嵌入障碍

我们将 $T^* \mathbb{T}^2- 0_{\mathbb{T}^2}$ 中的弱精确、有理拉格朗日环面分类为哈密顿同位素。这个结果与 $\mathbb{T}^n$ 上的闭 $1$-forms 的分类理论有关，也适用于辛拓扑。作为第一个推论，我们加强了由于 Eliashberg–Polterovich 和 Giroux 在 $T^* \mathbb{T}^2-0_{\mathbb{T}^2}$ 中描述拉格朗日圆环的独立结果，它们与零部分。作为第二个推论，我们展示了一对不相交的完全真实环面 $K_1, K_2 \subset T^*\mathbb{T}^2$，每个都是通过完全真实环面到零截面的同位素，但是这样并集$K_1 \cup K_2$ 甚至不是拉格朗日算子的平滑同位素。在论文的第二部分，我们研究了 $({\mathbb{R}}^4 中的拉格朗日圆环的链接，\omega )$ 和有理辛 $4$-流形。我们证明了这种环面的链接属性是由纯代数拓扑数据决定的，这通常可以从单调情况下的枚举磁盘计数中推断出来。我们还使用这个结果来描述某些拉格朗日嵌入障碍。