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Knot Floer homology of Whitehead doubles

Matthew Hedden

Geometry & Topology 11 (2007) 2277–2338

arXiv: math.GT/0606094


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.

Whitehead double, Heegaard diagram, Floer homology
Mathematical Subject Classification 2000
Primary: 57M27
Secondary: 57R58
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Received: 12 October 2006
Accepted: 20 August 2007
Published: 17 December 2007
Proposed: Ron Fintushel
Seconded: Peter Teichner, Ron Stern
Matthew Hedden
Department of Mathematics
Massachusetts Institute of Technology
77 Massachusetts Avenue
Cambridge MA 02139-4307