A note on the knot Floer homology of fibered knots

Abstract

We prove that the knot Floer homology of a fibered knot is nontrivial in its nextto-top Alexander grading. Immediate applications include new proofs of Krcatovich's result that knots with L-space surgeries are prime and Hedden and Watson's result that the rank of knot Floer homology detects the trefoil among knots in the 3-sphere. We also generalize the latter result, proving a similar theorem for nullhomologous knots in any 3-manifold. We note that our method of proof inspired Baldwin and Sivek's recent proof that Khovanov homology detects the trefoils. As part of this work, we also introduce a numerical refinement of the Ozsváth-Szabó contact invariant. This refinement was the inspiration for Hubbard and Saltz's annular refinement of Plamenevskaya's transverse link invariant in Khovanov homology.

Four-dimensional aspects of tight contact 3-manifolds

“…6 for its definition) is nontrivial. It is known that the contact manifolds with nonvanishing invariant form a proper subset of tight contact manifolds, and the extent to which the two classes differ is an interesting subject (7)(8)(9)(10)(11). One should compare our result to a Bennequin bound for surfaces in compact 4-manifolds with ∂W = Y and nontrivial Seiberg-Witten invariant relative to ξ, proved by Mrowka and…”

Khovanov homology detects the figure‐eight knot

“…Then is fibered, and we may assume that or else , which would contradict our conclusion that . Furthermore, we have that by [3], as above. In fact, since , we must have that .…”

“…It then follows from work of Baldwin and Vela‐Vick [3] that as well. (This result is stated in [3] with coefficients in but also holds over .) But this implies that , a contradiction.…”

“…Then is a fibered knot of genus [6, 11 18]. It then follows from work of Baldwin and Vela‐Vick [3] that as well. (This result is stated in [3] with coefficients in but also holds over .)…”

The next-to-top term in knot Floer homology

“…This inequality has been known for knots with thin knot Floer homology [10], L-space knots [4], fibered knots in any closed oriented 3-manifolds [1]. In this paper, we will prove (1) when HF K(Z, K, [F ], g) is supported in a single Z/2Zgrading.…”