Abstract
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.
Open book decompositions versus prime factorizations of closed, oriented 3-manifolds
