On the Colored Jones Polynomial, Sutured Floer Homology, and Knot Floer Homology

Let K in S3 be a knot, and let K denote the preimage of K inside its double branched cover, Sigma(S3, K). We prove, for each integer n \u3e 1, the existence of a spectral sequence from Khovanov\u27s categorification of the reduced n-colored Jones polynomial of K (mirror of K) and whose Einfinity term is the knot Floer homology of (Sigma(S3,K),K) (when n odd) and to (S3, K # K) (when n even). A corollary of our result is that Khovanov\u27s categorification of the reduced n-colored Jones polynomial detects the unknot whenever n\u3e1