Legendrian invariants and half Giroux torsion

Abstract: We collect some observations about Legendrian links with non-vanishing contact invariants, mostly concerning the non-loose realizations of links and the addition of boundary-parallel half Giroux torsion. In particular, we show that every null-homologous link with irreducible complement admits a non-loose Legendrian realization with non-zero (at least) invariant $\text{EH}$ in $\text{SFH}$, in some overtwisted contact structure (determined by its gradings); for many links these come from Gabai's work, for others the existence follows from the sutured interpretation of link Floer invariants. We reveal that separating half Giroux torsion does not necessary cause Legendrian invariants to vanish. Furthermore, we propose a conjectural characterization of links with non-vanishing $\widehat{\mathfrak L}$ in $\widehat{\text{HFL}}$ among links with non-zero $\mathfrak L$ in $c\text{HFL}^-$.