On the colored Jones polynomial, sutured Floer homology, and knot Floer homology

Let K⊂S3, and let denote the preimage of K inside its double branched cover, Σ(S3,K). We prove, for each integer n>1, the existence of a spectral sequence whose E2 term is Khovanov's categorification of the reduced n-colored Jones polynomial of (mirror of K) and whose E∞ term is the knot Floer homology of (when n odd) and of (S3,K#Kr) (when n even). A corollary of our result is that Khovanov's categorification of the reduced n-colored Jones polynomial detects the unknot whenever n>1.