Equivalence of two diagram representations of links in lens spaces and essential invariants

We study the relation between two diagrammatic representations of links in lens spaces: the disk diagram introduced in [8] and the grid diagram introduced in [2, 9] and we find how to shift from one to the other. We also investigate whether the HOMFLY-PT invariant and the Link Floer Homology are essential invariants, that is, we try to understand if these invariants are able to distinguish links in L(p, q) covered by the same link in S3. In order to do so, we generalize the combinatorial definition of Knot Floer Homology in lens spaces developed in [2, 19] to the case of links and we analyze how both the invariants change when we switch the orientation of the link.