Knot Floer homology of Whitehead doubles

Abstract

In this paper we study the knot Floer homology invariants of the twisted and untwisted Whitehead doubles of an arbitrary knot K. We present a formula for the filtered chain homotopy type of HFK(D(+,K,t)) in terms of the invariants for K, where D(+,K,t) denotes the t-twisted positive-clasped Whitehead double of K. In particular, the formula can be used iteratively and can be used to compute the Floer homology of manifolds obtained by surgery on Whitehead doubles. An immediate corollary is that tau(D(+,K,t))=1 if t< 2tau(K) and zero otherwise, where tau is the Ozsv{\'a}th-Szab{\'o} concordance invariant. It follows that the iterated untwisted Whitehead doubles of a knot satisfying tau(K)>0 are not smoothly slice.Comment: 41 pages, 14 color figures. spelling errors corrected and other minor change

Dehn surgery, rational open books and knot Floer homology

“…The key tool in this step is an understanding of the relationship between the knot Floer homology of a knot K ⊂ Y and the knot Floer homology of the core of the surgery torus K n ⊂ Y n . This relationship was studied in [He2], following the ideas of [OS5]. We begin this section with a detailed discussion of these results, and prove a generalization (Theorem 4.2) which will serve as the cornerstone of our proof.…”

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