Floer homology and non-fibered knot detection

TL;DR

This paper demonstrates that knot Floer homology, Khovanov homology, and HOMFLY homology can detect non-fibered knots, expanding their known capabilities from detecting only six fibered knots to infinitely many non-fibered ones.

Contribution

It provides the first proof that these homology theories can detect non-fibered knots and classifies genus-1 knots with specific Floer homology properties in the 3-sphere.

Findings

Knot Floer homology detects non-fibered knots.

HOMFLY homology detects infinitely many non-fibered knots.

Classification of genus-1 knots with 2-dimensional top Alexander grading Floer homology.

Abstract

We prove for the first time that knot Floer homology and Khovanov homology can detect non-fibered knots, and that HOMFLY homology detects infinitely many such knots; these theories were previously known to detect a mere six knots, all fibered. These results rely on our main technical theorem, which gives a complete classification of genus-1 knots in the 3-sphere whose knot Floer homology in the top Alexander grading is 2-dimensional. We discuss applications of this classification to problems in Dehn surgery which are carried out in two sequels. These include a proof that $0$-surgery characterizes infinitely many knots, generalizing results of Gabai from his 1987 resolution of the Property R Conjecture.