  We apply sutured Floer homology techniques to study the knot and link Floer
homologies of various links with annuli embedded in their exteriors. Our main
results include, for large $m$, characterizations of links with the same link
Floer homology as $(n,nm)$-cables of $L$-space knots or the same knot Floer
homology as $(2,2m)$-cables of $L$-space knots. Note that Knot Floer homology
carries less grading data than Link Floer homology, so the latter
characterizations are stronger than former. These characterizations yield some
new link detection results.
