We show that a quasipositive surface with disconnected boundary induces a map between the knot Floer homology groups of its boundary components preserving the transverse invariant. As an application, we show that this invariant can be used to obstruct decomposable Lagrangian cobordisms of arbitrary genus within Weinstein cobordisms. The construction of our maps rely on the comultiplicativity of the transverse invariant. Along the way, we also recover various naturality statements for the invariant under contact +1 surgery.1 decomposable Lagrangians in the symplectization of (S 3 , ξ std ) are built up from elementary cobordisms associated to births and pinches, we extend this notion to the setting of Weinstein cobordisms in Section 7.

We show that a decorated knot concordancewhich preserves the Alexander and absolute Z 2 -Maslov gradings. Our construction generalizes the concordance maps induced on HF K studied by Juhász and Marengon [JM2], but uses the description of HF K − as a direct limit of maps between sutured Floer homology groups discovered by Etnyre, Vela-Vick, and Zarev [EVZ].2010 Mathematics Subject Classification. 57M27; 57R58. Key words and phrases. concordance, contact, sutured, knot Floer homology, HF K − . 1 We use F = F 2 coefficients throughout this paper.

Suppose (B, π) is an open book supporting (Y, ξ), where the binding B is possibly disconnected, and K is a braid about this open book. Then B ∪ K is naturally a transverse link in (Y, ξ). We prove that the transverse link invariant in knot Floer homology,defined in [BVVV13] is always nonzero. This generalizes the main results of EVV10]. As an application, we show that if K is braided about an open book with connected binding, and has fractional Dehn twist coefficient greater than one, then t(K) = 0. This generalizes a result of Plamenevskaya [Pla15] for classical braids.

We discuss functoriality properties of the Ozsváth-Szabó contact invariant, and expose a number of results which seemed destined for folklore. We clarify the (in)dependence of the invariant on the basepoint, prove that it is functorial with respect to contactomorphisms, and show that it is strongly functorial under Stein cobordisms.

We discuss functoriality properties of the Ozsváth-Szabó contact invariant, and expose a number of results which seemed destined for folklore. We clarify the (in)dependence of the invariant on the basepoint, prove that it is functorial with respect to contactomorphisms, and show that it is strongly functorial under Stein cobordisms.