2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/171184Let K in S^3 be a knot, and let \widetilde{K} denote the preimage of K inside its double branched cover, Σ(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 the mirror of K to the knot Floer homology of (Σ(K),\widetilde{K}) (when n odd) and to (S^3, 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>1.46 pages, 13 figures; Unnecessary assumptions in statement of link surgeries spectral sequence (Section 4) removed, references updated, minor typos corrected throughoutGeometric TopologyQuantum AlgebraSymplectic Geometry57M27; 57R58; 57M12; 81R50text