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 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.