Knot Floer homology in cyclic branched covers

Abstract

In this paper, we introduce a sequence of invariants of a knot $K$ in $S^3$: the knot Floer homology groups $\widehat{HFK}\left({\Sigma }^m\left(K\right);\tilde{K},i\right)$ of the preimage of $K$ in the $m$–fold cyclic branched cover over $K$. We exhibit $\widehat{HFK}\left({\Sigma }^m\left(K\right);\tilde{K},i\right)$ as the categorification of a well-defined multiple of the Turaev torsion of ${\Sigma }^m\left(K\right)-\tilde{K}$ 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 $\widehat{HFK}\left({\Sigma }^2\left(K\right);\tilde{K},{\mathfrak{s}}_0\right)\cong \widehat{HFK}\left(S^3;K\right)$ 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 $\widehat{HFK}\left(S^3;K,i\right)$ for which $\widehat{HFK}\left({\Sigma }^2\left(K\right);\tilde{K},i\right)$ differ as ${\mathbb{Z}}_2$–graded groups.