On knot Floer homology and cabling

Abstract

This paper is devoted to the study of the knot Floer homology groups HF K(S 3 , K 2,n ), where K 2,n denotes the (2, n) cable of an arbitrary knot, K . It is shown that for sufficiently large |n|, the Floer homology of the cabled knot depends only on the filtered chain homotopy type of CF K(K). A precise formula for this relationship is presented. In fact, the homology groups in the top 2 filtration dimensions for the cabled knot are isomorphic to the original knot's Floer homology group in the top filtration dimension. The results are extended to (p, pn ± 1) cables. As an example we compute HF K((T 2,2m+1 ) 2,2n+1 ) for all sufficiently large |n|, where T 2,2m+1 denotes the (2, 2m + 1)-torus knot.

Rasmussen s-invariants of satellites do not detect slice knots

“…We remark that the behavior of τ under cabling is well understood by work of Hedden and Hom [Hed05,Hed09,Hom14a]. See also [VC10,Pet13].…”

Dehn surgery, rational open books and knot Floer homology

“…For the case where K is a knot in S 3 , this statement was established in [He1] (see also [Ni1] for a generalization to the case where Y is a rational homology sphere). However, the proof from [He1] and [Ni1] use Whitney disks to compare the Alexander gradings of different generators of CF (Y ); when b 1 (Y ) > 0, this proof no longer works since there may be no Whitney disks. Instead, we will use Lemma 2.1 to compare the gradings in the Heegaard diagrams of [He1,Ni1].…”

“…However, the proof from [He1] and [Ni1] use Whitney disks to compare the Alexander gradings of different generators of CF (Y ); when b 1 (Y ) > 0, this proof no longer works since there may be no Whitney disks. Instead, we will use Lemma 2.1 to compare the gradings in the Heegaard diagrams of [He1,Ni1]. Let F be a rational Seifert surface for K. We can find a doubly-pointed Heegaard diagram (Σ, α, β, w, z) for (Y, K), together with a longitude λ 0 on Σ and a relative periodic domain P representing F .…”

Knot Floer homology of Whitehead doubles

“…For a leisurely discussion of Heegaard diagrams, the two definitions and their equivalence, see Hedden [11]. Note, too, that we are thinking of knots which may not be embedded in the three-sphere, S 3 .…”

“…In fact, it will be more convenient to identify the domains of Whitney disks, by which we mean 2-chains lying in † gC2 with boundary in the attaching curves, and corner points contained in the .g C 2/-tuple of intersection points representing the generators. For the equivalence of these methods, see Hedden [11], Lipshitz [16] and Rasmussen [33]. Before beginning, recall the following definition, found in Ozsváth-Szabó [29].…”