Knot Floer homology and relative adjunction inequalities

Abstract

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

4-dimensional aspects of tight contact 3-manifolds

“…It generalizes the Ozsváth-Szabó-Rasmussen concordance invariant τ (K) for knots in S 3 and satisfies a Bennequin-type inequality analogous to one proved by Plamenevskaya in that setting [23]. The strategy will be to combine the Bennequin-bound for τ ξ (Y, K) with a relative adjunction inequality for the generalized τ invariants recently established by the authors in [14]. Combined, and correctly interpreted, these two inequalities will quickly yield Theorem 1 for surfaces with connected boundary; the extension to surfaces whose boundaries are multi-component Legendrian links will be deduced from properties of a "knotification" operation introduced by Ozsváth and Szabó [20, Section 2.1].…”

“…where K is any Legendrian representative of K in ξ. We wish to combine this inequality with a relative adjunction inequality for generalized τ invariants, recently proved by the authors in [14]. For this, define:…”

Relative genus bounds in indefinite four-manifolds

“…Invariants from Floer homology and Khovanov homology can be used to obstruct H-sliceness in definite four-manifolds [48,36,39,24].…”

“…Remark 4.9. Theorem 1.1 should be compared to another relative adjunction inequality, due to Hedden and Raoux [24]. They proved that, if X be a smooth, oriented four-manifold with boundary Y and Σ ⊂ X a properly smoothly embedded surface such that the relative element…”

“…When X = S 4 , the problem reduces to the well-known question of finding the four-ball genus of knots, and in particular of determining which knots are slice. More generally, when X has definite intersection form, many of the gauge theoretic techniques for bounding the genus of embedded surfaces still apply; see [48,36,24]. When X = # n CP 2 or # n CP 2 , there are also bounds from Khovanov homology [39].…”