Knot Floer homology in cyclic branched covers

In this paper, we introduce a sequence of invariants of a knot K in S3 : the knot Floer homology groups ĤFK(Σm(K); K̃, i) of the preimage of K in the m–fold cyclic branched cover over K . We exhibit ĤFK(Σm(K); K̃, i) as the categorification of a well-defined multiple of the Turaev torsion of Σm(K)− K̃ in the case where Σm(K) is a rational homology sphere. In addition, when K is a two-bridge knot, we prove that ĤFK(Σ2(K); K̃, s0) ∼= ĤFK(S3; K) for s0 the spin Spinc structure on Σ2(K). We conclude with a calculation involving two knots with identical ĤFK(S3; K, i) for which ĤFK(Σ2(K); K̃, i) differ as Z2 –graded groups.