Knot Floer homology, link Floer homology and link detection

We give new link detection results for knot and link Floer homology, inspired by recent work on Khovanov homology.We show that knot Floer homology detects T .2;4/, T .2;6/, T .3;3/, L7n1 and the link T .2;2n/ with the orientation of one component reversed.We show link Floer homology detects T .2;2n/ and T .n;n/, for all n.Additionally, we identify infinitely many pairs of links such that both links in the pair are each detected by link Floer homology but have the same Khovanov homology and knot Floer homology.Finally, we use some of our knot Floer detection results to give topological applications of annular Khovanov homology. 57K10, 57K181 Introduction Knot and link Floer homology are invariants of links in S 3 ; see Ozsváth and Szabó 31;32 and Rasmussen 34.There are a number of formal similarities between these Floer theoretic invariants and the combinatorial Khovanov homology.Recently, Khovanov homology has been shown to detect a number of simple links; see Baldwin, Dowlin, Levine, Lidman and Sazdanovic 2, Li, Xie and Zhang 23, Martin 26 and Xie and Zhang 38;40.Some of these detection results have used knot and link Floer homology without going so far as to determine whether knot or link Floer homology detects the relevant link.Inspired by this work, we give such detection results for knot and link Floer homology.We remind the reader that the knot Floer homology of a link L is computed using an associated knot, called the knotification of L, in a connected sum of copies of S 1 S 2 , while the link Floer homology of L is computed directly from the link L in S 3 .Previously it was known that knot Floer homology detects the unknot (see Ozsváth and Szabó 30), the trefoil (see Ghiggini 7), the figure eight knot 7, the Hopf link (see Ni 28 and 30) and the unlink (see Hedden and Watson 15 and Ni 29).Link Floer homology was known to detect the trivial n-braid together with its braid axis (see Baldwin and Grigsby 3) and determine if a link is split; see Wang 37.It was also known that a stronger version of link Floer homology, CFL 1 , detects the Borromean rings and the Whitehead link; see Gorsky, Lidman, B Liu and Moore 11.We prove the following knot Floer homology detection results: Theorem 4.1 If b HFK.L/ Š b