Geometry & Topology

Knot Floer homology of Whitehead doubles

Matthew Hedden

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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 HFK̂(D±(K,t)) in terms of the invariants for K, where D±(K,t) 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 τ(D+(K,t))=1 if t<2τ(K) and zero otherwise, where τ is the Ozsváth–Szabó concordance invariant. It follows that the iterated untwisted Whitehead doubles of a knot satisfying τ(K)>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.

Article information

Geom. Topol., Volume 11, Number 4 (2007), 2277-2338.

Received: 12 October 2006
Accepted: 20 August 2007
First available in Project Euclid: 20 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M27: Invariants of knots and 3-manifolds
Secondary: 57R58: Floer homology

Whitehead double Heegaard diagram Floer homology


Hedden, Matthew. Knot Floer homology of Whitehead doubles. Geom. Topol. 11 (2007), no. 4, 2277--2338. doi:10.2140/gt.2007.11.2277.

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