A note on the knot Floer homology of fibered knots

Baldwin, John; Vela-Vick, David

doi:10.2140/agt.2018.18.3669

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A note on the knot Floer homology of fibered knots

John A Baldwin and David Shea Vela-Vick

Algebraic & Geometric Topology 18 (2018) 3669–3690

DOI: 10.2140/agt.2018.18.3669

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.

Keywords

open book, transverse braid, Heegaard Floer homology

Mathematical Subject Classification 2010

Primary: 57M27

Secondary: 57R17, 57R58

References

Publication

Received: 19 January 2018

Revised: 6 July 2018

Accepted: 17 July 2018

Published: 18 October 2018

Authors

John A Baldwin
Department of Mathematics Boston College Chestnut Hill, MA United States
https://www2.bc.edu/john-baldwin/
David Shea Vela-Vick
Department of Mathematics Louisiana State University Baton Rouge, LA United States
http://www.math.lsu.edu/~shea

Volume 18, issue 6 (2018)