# Khovanov homology is an unknot-detector

@article{Kronheimer2010KhovanovHI, title={Khovanov homology is an unknot-detector}, author={Peter B. Kronheimer and Tomasz S. Mrowka}, journal={Publications math{\'e}matiques de l'IH{\'E}S}, year={2010}, volume={113}, pages={97-208} }

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 cohomology and abutting to a knot homology defined using singular instantons. We then show that the latter homology is isomorphic to the instanton Floer homology of the sutured knot complement: an invariant that is already known to detect the unknot.

## Figures from this paper

## 219 Citations

Khovanov homology and the cinquefoil

- Mathematics
- 2021

We prove that Khovanov homology with coefficients in Z/2Z detects the (2, 5) torus knot. Our proof makes use of a wide range of deep tools in Floer homology, Khovanov homology, and Khovanov homotopy.…

Unknotting number and Khovanov homology

- MathematicsPacific Journal of Mathematics
- 2019

We show that the order of torsion homology classes in Bar-Natan deformation of Khovanov homology is a lower bound for the unknotting number. We give examples of knots that this is a better lower…

Khovanov homology and rational unknotting

- Mathematics
- 2021

Building on work by Alishahi-Dowlin, we extract a new knot invariant λ ≥ 0 from universal Khovanov homology. While λ is a lower bound for the unknotting number, in fact more is true: λ is a lower…

Khovanov homology detects the trefoils

- Mathematics
- 2018

We prove that Khovanov homology detects the trefoils. Our proof incorporates an array of ideas in Floer homology and contact geometry. It uses open books; the contact invariants we defined in the…

An untwisted cube of resolutions for knot Floer homology

- Mathematics
- 2011

Ozsvath and Szabo gave a combinatorial description of knot Floer homology based on a cube of resolutions, which uses maps with twisted coefficients. We study the t=1 specialization of their…

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…

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…

A link-splitting spectral sequence in Khovanov homology

- Mathematics
- 2015

We construct a new spectral sequence beginning at the Khovanov homology of a link and converging to the Khovanov homology of the disjoint union of its components. The page at which the sequence…

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.…

Plumbing essential states in Khovanov homology

- 2018

We prove that every homogeneously adequate Kauffman state has enhancements X± in distinct j-gradings whose traces (which we define) represent nonzero Khovanov homology classes over Z/2Z; this is also…

## References

SHOWING 1-10 OF 40 REFERENCES

Khovanov homology and the slice genus

- Mathematics
- 2004

We use Lee’s work on the Khovanov homology to define a knot invariant s. We show that s(K) is a concordance invariant and that it provides a lower bound for the smooth slice genus of K. As a…

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

- Mathematics
- 2008

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…

Instanton Floer homology and the Alexander polynomial

- Mathematics
- 2010

The instanton Floer homology of a knot in the three-sphere is a vector space with a canonical mod 2 grading. It carries a distinguished endomorphism of even degree,arising from the 2-dimensional…

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 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…

An unoriented skein exact triangle for knot Floer homology

- Mathematics
- 2006

Given a crossing in a planar diagram of a link in the three-sphere, we show that the knot Floer homologies of the link and its two resolutions at that crossing are related by an exact triangle. As a…

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…

Monopoles and lens space surgeries

- Mathematics
- 2003

Monopole Floer homology is used to prove that real projective three-space cannot be obtained from Dehn surgery on a nontrivial knot in the three-sphere. To obtain this result, we use a surgery long…

Floer homology and knot complements

- Mathematics
- 2003

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…

On the spectral sequence from Khovanov homology to Heegaard Floer homology

- Mathematics
- 2008

Ozsvath and Szabo show that there is a spectral sequence whose E^2 term is the reduced Khovanov homology of L, and which converges to the Heegaard Floer homology of the (orientation reversed)…