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

Details On the Colored Jones Polynomial, Sutured Floer homology, and Knot Floer homology On the Colored Jones Polynomial, Sutured Floer homology, and Knot Floer homology

2008-07-09

arxiv.org/abs/0807.1432v3

Mathematics

Geometric Topology

46 pages, 13 figures; Unnecessary assumptions in statement of link surgeries spectral sequence (Section 4) removed, references

Scientific paper

Let K in S^3 be a knot, and let \widetilde{K} denote the preimage of K inside its double branched cover, \Sigma(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 (\Sigma(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.

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