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
–space
surgeries are prime and Hedden and Watson’s result that the rank
of knot Floer homology detects the trefoil among knots in the
–sphere. We
also generalize the latter result, proving a similar theorem for nullhomologous knots in any
–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.
Keywords
open book, transverse braid, Heegaard Floer homology
