On knot Floer homology and cabling

Abstract

This paper is devoted to the study of the knot Floer homology groups $\widehat{HFK}\left(S^3,K_{2,n}\right)$, where $K_{2,n}$ denotes the $\left(2,n\right)$ cable of an arbitrary knot, $K$. It is shown that for sufficiently large $\left|n\right|$, the Floer homology of the cabled knot depends only on the filtered chain homotopy type of $\widehat{CFK}\left(K\right)$. 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 $\left(p,pn\pm 1\right)$ cables. As an example we compute $\widehat{HFK}\left({\left(T_{2,2m+1}\right)}_{2,2n+1}\right)$ for all sufficiently large $\left|n\right|$, where $T_{2,2m+1}$ denotes the $\left(2,2m+1\right)$–torus knot.