Geometry & Topology

Knot Floer homology of Whitehead doubles

Matthew Hedden

Full-text: Open access

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 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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513799983

Digital Object Identifier
doi:10.2140/gt.2007.11.2277

Mathematical Reviews number (MathSciNet)
MR2372849

Zentralblatt MATH identifier
1187.57015

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

Keywords
Whitehead double Heegaard diagram Floer homology

Citation

Hedden, Matthew. Knot Floer homology of Whitehead doubles. Geom. Topol. 11 (2007), no. 4, 2277--2338. doi:10.2140/gt.2007.11.2277. https://projecteuclid.org/euclid.gt/1513799983


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