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 > 1, the existence of a spectral sequence from Khovanov's 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's categorification of the reduced n-colored Jones polynomial detects the unknot whenever n>1.
Grigsby, J. Elisenda and Wehrli, Stephan, "On the Colored Jones Polynomial, Sutured Floer Homology, and Knot Floer Homology" (2008). Mathematics - All Scholarship. 115.
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