# Khovanov homology of the \$2\$-cable detects the unknot

```@article{Hedden2008KhovanovHO,
title={Khovanov homology of the \\$2\\$-cable detects the unknot},
author={Matthew Hedden},
journal={Mathematical Research Letters},
year={2008},
volume={16},
pages={991-994}
}```
• M. Hedden
• Published 28 May 2008
• Mathematics
• Mathematical Research Letters
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 detects the unknot. A corollary is that Khovanov's categorification of the 2-colored Jones polynomial detects the unknot.
On Gradings in Khovanov homology and sutured Floer homology
• Mathematics
• 2010
We discuss generalizations of Ozsvath-Szabo's spectral sequence relating Khovanov homology and Heegaard Floer homology, focusing attention on an explicit relationship between natural Z (resp., 1/2 Z)
Khovanov module and the detection of unlinks
• Mathematics
• 2013
We study a module structure on Khovanov homology, which we show is natural under the Ozsvath–Szabo spectral sequence to the Floer homology of the branched double cover. As an application, we show
Khovanov homology is an unknot-detector
• Mathematics
• 2010
We prove that a knot is the unknot if and only if its reduced Khovanov cohomology has rank 1. The proof has two steps. We show first that there is a spectral sequence beginning with the reduced
Khovanov homology also detects split links
• Mathematics
• 2019
Extending ideas of Hedden-Ni, we show that the module structure on Khovanov homology detects split links. We also prove an analogue for untwisted Heegaard Floer homology of the branched double cover.
Reduced 2-coloured Khovanov Homology detects the Trefoil
We prove that the reduced 2-coloured Khovanov homology detects the trefoil, using a spectral sequence to knot Floer homology.
Does Khovanov homology detect the unknot?
• Mathematics
• 2008
<abstract abstract-type="TeX"><p> We determine a class of knots, which includes unknotting number one knots, within which Khovanov homology detects the unknot. A corollary is that the Khovanov
Manifolds with small Heegaard Floer ranks
• Mathematics
• 2010
We show that the only irreducible three-manifold with positive first Betti number and Heegaard Floer homology of rank two is homeomorphic to zero-framed surgery on the trefoil. We classify links
State cycles, quasipositive modification, and constructing H-thick knots in Khovanov homology
We study Khovanov homology classes which have state cycle representatives, and examine how they interact with Jacobsson homomorphisms and Lee's map \$\Phi\$. As an application, we describe a general
A HITCHHIKER'S GUIDE TO KHOVANOV HOMOLOGY
These notes from the 2014 summer school Quantum Topology at the CIRM in Luminy attempt to provide a rough guide to a selection of developments in Khovanov homology over the last fifteen years.
On the Colored Jones Polynomial, Sutured Floer homology, and Knot Floer homology
• Mathematics
• 2008
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

## References

SHOWING 1-10 OF 25 REFERENCES
Does Khovanov homology detect the unknot?
• Mathematics
• 2008
<abstract abstract-type="TeX"><p> We determine a class of knots, which includes unknotting number one knots, within which Khovanov homology detects the unknot. A corollary is that the Khovanov
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
A categorification of the Jones polynomial
Author(s): Khovanov, Mikhail | Abstract: We construct a bigraded cohomology theory of links whose Euler characteristic is the Jones polynomial.
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
Torsion of the Khovanov homology
Khovanov homology is a recently introduced invariant of oriented links in \$\mathbb{R}^3\$. It categorifies the Jones polynomial in the sense that the (graded) Euler characteristic of the Khovanov
On the Colored Jones Polynomial, Sutured Floer homology, and Knot Floer homology
• Mathematics
• 2008
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
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
Link homology and categorification
This is a short survey of algebro-combinatorial link homology theories which have the Jones polynomial and other link polynomials as their Euler characteristics.
Holomorphic disks and genus bounds
• Mathematics
• 2004
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
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