On knot Floer homology and cabling

Details On knot Floer homology and cabling On knot Floer homology and cabling

2004-06-21

arxiv.org/abs/math/0406402v2

Algebr. Geom. Topol. 5 (2005) 1197-1222

Mathematics

Geometric Topology

Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-48.abs.html

Scientific paper

This paper is devoted to the study of the knot Floer homology groups HFK(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 CFK(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,pn+-1) cables. As an example we compute HFK((T_{2,2m+1})_{2,2n+1}) for all sufficiently large |n|, where T_{2,2m+1} denotes the (2,2m+1)-torus knot.

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