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

```@article{Grigsby2008OnTC,
title={On the Colored Jones Polynomial, Sutured Floer homology, and Knot Floer homology},
author={Julia Elisenda Grigsby and Stephan Martin Wehrli},
journal={arXiv: Geometric Topology},
year={2008}
}```
• Published 9 July 2008
• Mathematics
• arXiv: Geometric Topology
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…

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## References

SHOWING 1-10 OF 26 REFERENCES
Floer homology and knot complements
We use the Ozsvath-Szabo theory of Floer homology to define an invariant of knot complements in three-manifolds. This invariant takes the form of a filtered chain complex, which we call CF_r. It
Floer homology and surface decompositions
Sutured Floer homology, denoted by SFH, is an invariant of balanced sutured manifolds previously defined by the author. In this paper we give a formula that shows how this invariant changes under
On the Heegaard Floer homology of branched double-covers
• Mathematics
• 2003
Abstract Let L ⊂ S 3 be a link. We study the Heegaard Floer homology of the branched double-cover Σ ( L ) of S 3 , branched along L. When L is an alternating link, HF ^ of its branched double-cover
Link Floer homology detects the Thurston norm
We prove that, for a link L in a rational homology 3–sphere, the link Floer homology detects the Thurston norm of its complement. This result has been proved by Ozsvath and Szabo for links in S^3. As
Holomorphic disks and knot invariants
• Mathematics
• 2002
Abstract We define a Floer-homology invariant for knots in an oriented three-manifold, closely related to the Heegaard Floer homologies for three-manifolds defined in an earlier paper. We set up
On the Khovanov and knot Floer homologies of quasi-alternating links
• Mathematics
• 2007
Quasi-alternating links are a natural generalization of alternating links. In this paper, we show that quasi-alternating links are "homologically thin" for both Khovanov homology and knot Floer
Categorification of the Kauffman bracket skein module of I-bundles over surfaces
• Mathematics
• 2004
Khovanov defined graded homology groups for links LR 3 and showed that their polynomial Euler characteristic is the Jones polyno- mial of L. Khovanov's construction does not extend in a
Holomorphic disks and genus bounds
We prove that, like the Seiberg{Witten monopole homology, the Heegaard Floer homology for a three-manifold determines its Thurston norm. As a consequence, we show that knot Floer homology detects the
Khovanov homology of the \$2\$-cable detects the unknot
Inspired by recent work of Grigsby and Werhli, we use the deep geometric content of Ozsvath and Szabo's Floer homology theory to provide a short proof that the Khovanov homology of the 2-cable
Categorifications of the colored Jones polynomial
The colored Jones polynomial of links has two natural normalizations: one in which the n-colored unknot evaluates to [n+1], the quantum dimension of the (n+1)-dimensional irreducible representation