On knot Floer homology and cabling

This paper is devoted to the study of the knot Floer homology groups $\hat{H F K} (S^{3}, K_{2, n})$ , where $K_{2, n}$ denotes the $(2, n)$ cable of an arbitrary knot, $K$ . It is shown that for sufficiently large $| n |$ , the Floer homology of the cabled knot depends only on the filtered chain homotopy type of $\hat{C F K} (K)$ . A precise formula for this relationship is presented. In fact, the homology groups in the top $2$ filtration dimensions for the cabled knot are isomorphic to the original knot’s Floer homology group in the top filtration dimension. The results are extended to $(p, p n \pm 1)$ cables. As an example we compute $\hat{H F K} ({(T_{2, 2 m + 1})}_{2, 2 n + 1})$ for all sufficiently large $| n |$ , where $T_{2, 2 m + 1}$ denotes the $(2, 2 m + 1)$ –torus knot.

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Keywords

knots, Floer homology, cable, satellite, Heegaard diagrams

Mathematical Subject Classification 2000

Primary: 57M27

Secondary: 57R58

References

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Publication

Received: 9 August 2004

Revised: 23 July 2005

Accepted: 14 March 2005

Published: 20 September 2005

Authors

Matthew Hedden
Department of Mathematics Princeton University Princeton NJ 08544-1000 USA

Volume 5, issue 3 (2005)

Matthew Hedden

Abstract