With a 4‐ended tangle $T$, we associate a Heegaard Floer invariant ${CFT}^{\partial }(T)$, the peculiar module of $T$. Based on Zarev's bordered sutured Heegaard Floer theory (Zarev, PhD Thesis, Columbia University, 2011), we prove a glueing formula for this invariant which recovers link Floer homology $\widehat{HFL}$. Moreover, we classify peculiar modules in terms of immersed curves on the 4‐punctured sphere. In fact, based on an algorithm of Hanselman, Rasmussen and Watson (Preprint, 2016, arXiv:1604.03466v2), we prove general classification results for the category of curved complexes over a marked surface with arc system. This allows us to reinterpret the glueing formula for peculiar modules in terms of Lagrangian intersection Floer theory on the 4‐punctured sphere.

中文翻译：

---

用于四端缠结的特殊模块

带有四端纠缠 $Ť$，我们将Heegaard Floer不变 ${CFT}^{\partial }（Ť）$，是 $Ť$。基于Zarev的边界缝合Heegaard Floer理论（Zarev，哥伦比亚大学博士学位论文，2011年），我们证明了该不变式的胶合公式，该公式恢复了链接Floer的同源性$\widehat{HF\mathrm{大号}}$。此外，我们根据4针球体上的浸入曲线对特殊模块进行分类。实际上，基于Hanselman，Rasmussen和Watson的算法（Preprint，2016，arXiv：1604.03466v2），我们证明了带有弧系统的标记表面上的曲面复合体类别的一般分类结果。这使我们可以根据四点球上的拉格朗日相交弗洛尔理论重新解释特殊模块的粘合公式。