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, $K$. A formula is presented for the filtered chain homotopy type of $\widehat{HFK}\left(D_{\pm }\left(K,t\right)\right)$ in terms of the invariants for $K$, where $D_{\pm }\left(K,t\right)$ denotes the $t$–twisted positive (resp. negative)-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 $\tau \left(D_{+}\left(K,t\right)\right)=1$ if $t<2\tau \left(K\right)$ and zero otherwise, where $\tau$ is the Ozsváth–Szabó concordance invariant. It follows that the iterated untwisted Whitehead doubles of a knot satisfying $\tau \left(K\right)>0$ are not smoothly slice. Another corollary is a closed formula for the Floer homology of the three-manifold obtained by gluing the complement of an arbitrary knot, $K$, to the complement of the trefoil.