Concordance maps in knot Floer homology

Abstract

We modify the construction of knot Floer homology to produce a one-parameter family of homologies tHFK for knots in S 3. These invariants can be used to give homomorphisms from the smooth concordance group C to Z, giving bounds on the four-ball genus and the concordance genus of knots. We give some applications of these homomorphisms.

Link cobordisms and functoriality in link Floer homology

“…t . Using this fact, it is straightforward to see that (after perhaps a small perturbation of the path f t ) the parametrized Kirby decompositions K(f a , b a ) and (10), Move (11), Move (13) or Move (14).…”

Distinguishing slice disks using knot Floer homology

“…On the other hand, t D C corresponds to F C by Theorem 6.1. As C is invertible, the map F C is injective, and hence has rank dim(V ); see [JM16]. But K = U , so dim(V ) > 1 by the genus detection of knot Floer homology.…”

“…Kim [Kim10] has shown that every knot K admits an invertible concordance C to a prime knot K , obtained by taking a certain satellite of K. Let P and P be decorations on K and K , respectively, choose a decoration σ on C compatible with these, and let C = (C, σ). If D and D are slice disks of K with t D,P = t D ,P , then t C∪D,P = t C∪D ,P , since t C∪D,P = F C (t D,P ) and t C∪D ,P = F C (t D ,P ), and the concordance map F C is injective; see [JM16]. In other words, if the invariant distinguishes the slice disks D and D of a possibly composite knot K, then it also distinguishes the slice disks C ∪ D and C ∪ D of the prime knot K , up to stable isotopy.…”

Link cobordisms and absolute gradings on link Floer homology

“…We note that in [JM16] and [JM18], Juhász and Marengon compute the Alexander and Maslov grading changes for Juhász's link cobordism maps on HFL when the underlying 4-manifold is [0, 1] × S 3 . Their formula for the grading changes agree with the ones from Theorem 1.4 (though in their case, the only non-zero terms involve the Euler characteristic of the subsurfaces).…”