#### Volume 6, issue 3 (2006)

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Knot Floer homology in cyclic branched covers

### J Elisenda Grigsby

Algebraic & Geometric Topology 6 (2006) 1355–1398
 arXiv: math.GT/0507498
##### Abstract

In this paper, we introduce a sequence of invariants of a knot $K$ in ${S}^{3}$: the knot Floer homology groups $\stackrel{̂}{HFK}\left({\Sigma }^{m}\left(K\right);\stackrel{˜}{K},i\right)$ of the preimage of $K$ in the $m$–fold cyclic branched cover over $K$. We exhibit $\stackrel{̂}{HFK}\left({\Sigma }^{m}\left(K\right);\stackrel{˜}{K},i\right)$ as the categorification of a well-defined multiple of the Turaev torsion of ${\Sigma }^{m}\left(K\right)-\stackrel{˜}{K}$ in the case where ${\Sigma }^{m}\left(K\right)$ is a rational homology sphere. In addition, when $K$ is a two-bridge knot, we prove that $\stackrel{̂}{HFK}\left({\Sigma }^{2}\left(K\right);\stackrel{˜}{K},{\mathfrak{s}}_{0}\right)\cong \stackrel{̂}{HFK}\left({S}^{3};K\right)$ for ${\mathfrak{s}}_{0}$ the spin Spin${}^{c}$ structure on ${\Sigma }^{2}\left(K\right)$. We conclude with a calculation involving two knots with identical $\stackrel{̂}{HFK}\left({S}^{3};K,i\right)$ for which $\stackrel{̂}{HFK}\left({\Sigma }^{2}\left(K\right);\stackrel{˜}{K},i\right)$ differ as ${ℤ}_{2}$–graded groups.

##### Keywords
Heegaard Floer homology, branched covers
##### Mathematical Subject Classification 2000
Primary: 57R58, 57M27
Secondary: 57M05