#### Volume 5, issue 3 (2005)

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On knot Floer homology and cabling

### Matthew Hedden

Algebraic & Geometric Topology 5 (2005) 1197–1222
 arXiv: math.GT/0406402
##### Abstract

This paper is devoted to the study of the knot Floer homology groups $\stackrel{̂}{HFK}\left({S}^{3},{K}_{2,n}\right)$, where ${K}_{2,n}$ denotes the $\left(2,n\right)$ 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 $\stackrel{̂}{CFK}\left(K\right)$. 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 $\left(p,pn±1\right)$ cables. As an example we compute $\stackrel{̂}{HFK}\left({\left({T}_{2,2m+1}\right)}_{2,2n+1}\right)$ for all sufficiently large $|n|$, where ${T}_{2,2m+1}$ denotes the $\left(2,2m+1\right)$–torus knot.

##### Keywords
knots, Floer homology, cable, satellite, Heegaard diagrams
Primary: 57M27
Secondary: 57R58