#### Volume 11, issue 4 (2007)

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Knot Floer homology of Whitehead doubles

### Matthew Hedden

Geometry & Topology 11 (2007) 2277–2338
 arXiv: math.GT/0606094
##### 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 $\stackrel{̂}{HFK}\left({D}_{±}\left(K,t\right)\right)$ in terms of the invariants for $K$, where ${D}_{±}\left(K,t\right)$ 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.

##### Keywords
Whitehead double, Heegaard diagram, Floer homology
Primary: 57M27
Secondary: 57R58