### Abstract

Let K ⊂ S^{3}, and let over(K, ̃) denote the preimage of K inside its double branched cover, Σ (S^{3}, K). We prove, for each integer n > 1, the existence of a spectral sequence whose E^{2} term is Khovanov's categorification of the reduced n-colored Jones polynomial of over(K, -) (mirror of K) and whose E^{∞} term is the knot Floer homology of (Σ (S^{3}, K), over(K, ̃)) (when n odd) and of (S^{3}, K # K^{r}) (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.

Original language | English (US) |
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Pages (from-to) | 2114-2165 |

Number of pages | 52 |

Journal | Advances in Mathematics |

Volume | 223 |

Issue number | 6 |

DOIs | |

State | Published - Apr 1 2010 |

### Keywords

- Colored Jones polynomial
- Floer homology
- Khovanov homology
- Link invariants
- Sutured manifolds
- Unknot detection

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

Elisenda Grigsby, J., & Wehrli, S. M. (2010). On the colored Jones polynomial, sutured Floer homology, and knot Floer homology.

*Advances in Mathematics*,*223*(6), 2114-2165. https://doi.org/10.1016/j.aim.2009.11.002