Knot Floer homology of Whitehead doubles

Hedden, Matthew

doi:10.2140/gt.2007.11.2277

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Knot Floer homology of Whitehead doubles

Matthew Hedden

Geometry & Topology 11 (2007) 2277–2338

DOI: 10.2140/gt.2007.11.2277

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 $\hat{H F K} (D_{\pm} (K, t))$ in terms of the invariants for $K$ , where $D_{\pm} (K, t)$ 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 $t < 2 τ (K)$ 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.

Keywords

Whitehead double, Heegaard diagram, Floer homology

Mathematical Subject Classification 2000

Primary: 57M27

Secondary: 57R58

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Publication

Received: 12 October 2006

Accepted: 20 August 2007

Published: 17 December 2007

Proposed: Ron Fintushel

Seconded: Peter Teichner, Ron Stern

Authors

Matthew Hedden
Department of Mathematics Massachusetts Institute of Technology 77 Massachusetts Avenue Cambridge MA 02139-4307 USA
http://math.mit.edu/~mhedden

Volume 11, issue 4 (2007)