We introduce a generalization of the Lisca–Ozsváth–Stipsicz–Szabó Legendrian invariant ${\mathfrak L}$ to links in every rational homology sphere, using the collapsed version of link Floer homology. We represent a Legendrian link L in a contact 3-manifold ${(M,\xi)}$ with a diagram D, given by an open book decomposition of ${(M,\xi)}$ adapted to L, and we construct a chain complex ${cCFL^-(D)}$ with a special cycle in it denoted by ${\mathfrak L(D)}$ . Then, given two diagrams ${D_1}$ and ${D_2}$ which represent Legendrian isotopic links, we prove that there is a map between the corresponding chain complexes that induces an isomorphism in homology and sends ${\mathfrak L(D_1)}$ into ${\mathfrak L(D_2)}$ . Moreover, a connected sum formula is also proved and we use it to give some applications about non-loose Legendrian links; that are links such that the restriction of ${\xi}$ on their complement is tight.

中文翻译：

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接触三流形的开书分解的传奇和横向链接的不变量

我们介绍了 Lisca–Ozsváth–Stipsicz–Szabó Legendrian 不变量的推广${\mathfrak L}$使用链接 Floer 同调的折叠版本来链接每个有理同调领域。我们代表一个传奇的链接大号在接触 3 歧管中${(M,\xi)}$带图D，由打开的书分解给出${(M,\xi)}$适应大号，我们构造了一个链复形${cCFL^-(D)}$其中有一个特殊的循环，表示为${\mathfrak L(D)}$. 然后给定两张图${D_1}$和${D_2}$代表Legendrian同位素链接，我们证明在相应的链配合物之间存在一个映射，该映射在同源性中引起同构并发送${\mathfrak L(D_1)}$进入${\mathfrak L(D_2)}$. 此外，还证明了一个连通和公式，并用它给出了关于非松散Legendrian链接的一些应用；这些链接使得限制${\xi}$对他们的补充很紧。