A characterization of contact elements

We show that for a non-trivial element c in HF(-Y), there exists a tight contact structure ξ on Y whose contact invariant realizes c if and only if there exists a non-trivial fibered knot K such that -τc(K)=g(K). Moreover, when such a fibered knot K does exist, ξ can be chosen to satisfy the extra condition that K admits a Legendrian representative with Thurston-Bennequin number tb$(K)$ equal to $0$ in (Y,ξ).