AN INVARIANT OF LEGENDRIAN AND TRANSVERSE LINKS FROM OPEN BOOK DECOMPOSITIONS OF CONTACT 3-MANIFOLDS

CAVALLO, ALBERTO

Glasgow mathematical journal, Tome 63 (2021) no. 2, pp. 451-483

Résumé

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.

Détail

DOI : 10.1017/S0017089520000300