# 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} }

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.

## 19 Citations

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

- Mathematics
- 2018

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

- Mathematics
- 2010

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

- Physics
- 2014

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.

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Does Khovanov homology detect the unknot?

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<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…

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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…

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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…