A note on the knot Floer homology of fibered knots

TL;DR

This paper proves that the knot Floer homology of fibered knots is nontrivial in a specific Alexander grading, leading to new proofs of existing results and generalizations, and introduces a numerical refinement of the contact invariant.

Contribution

It establishes nontriviality of knot Floer homology in a key grading for fibered knots and introduces a new numerical contact invariant refinement.

Findings

Nontrivial in next-to-top Alexander grading for fibered knots

New proofs that L-space surgeries imply knots are prime

Generalization of detection results to knots in any 3-manifold

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 trefoils. As part of this work, we also introduce a numerical refinement of the Ozsv\'ath-Szab\'o contact invariant. This refinement was the inspiration for Hubbard and Saltz's annular refinement of Plamenevskaya's transverse link invariant in Khovanov homology.