Knot Floer homology and relative adjunction inequalities

We establish inequalities that constrain the genera of smooth cobordisms between knots in 4-dimensional cobordisms. These “relative adjunction inequalities” improve for closed surfaces which have been instrumental many topological applications gauge theory. The relative refine latter by incorporating numerical invariants boundary associated to Heegaard Floer homology classes determined 4-manifold. As a corollary, we produce host concordance general 3-manifold, one such invariant every non-zero class. apply our results analogues Ozsváth–Szabó–Rasmussen links, allowing us reprove link version Milnor conjecture, and, furthermore, show knot detects strongly quasipositive fibered links.