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 next-to-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 trefoil. 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.