Martin showed that link Floer homology detects braid axes. In this paper we extend this result to give a topological characterisation of links which are almost braided from the point of view of link Floer homology. The result is inspired by work of Baldwin-Sivek and Li-Ye on nearly fibered knots. Applications include that Khovanov homology detects the Whitehead link and $L7n2$, as well as infinite families of detection results for link Floer homology and annular Khovanov homology.
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Binns et al. (2024) studied this question.
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