## Khovanov homology is an unknot-detector

Citations: | 4 - 2 self |

### BibTeX

@MISC{Kronheimer_khovanovhomology,

author = {P. B. Kronheimer and T. S. Mrowka},

title = {Khovanov homology is an unknot-detector},

year = {}

}

### OpenURL

### Abstract

Abstract. 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. 1

### Citations

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The Geometry of Four-Manifolds
- Donaldson, Kronheimer
- 1990
(Show Context)
Citation Context ... is a unique anti-selfdual SO.3/ connection with D 1=4. This connection A 2 CP N is reducible and has non-zero w2: it splits as R ˚ L, where L is an oriented 2-plane bundle with e.L/ŒCP 1 � D 1. (See =-=[7]-=- for example.) We can view L as the tautological line bundle on CP N 2 , and as such we see that the action of complex conjugation on CP N 2 lifts to an involution on L that is orientation-reversing o... |

260 |
A categorification of the Jones polynomial
- Khovanov
- 2000
(Show Context)
Citation Context ...omplement: an invariant that is already known to detect the unknot. 1 Introduction 1.1 Statement of results This paper explores a relationship between the Khovanov cohomology of a knot, as defined in =-=[16]-=-, and various homology theories defined using Yang-Mills instantons, of which the archetype is Floer’s instanton homology of 3-manifolds [8]. A consequence of this relationship is a proof that Khovano... |

116 | Floer homology and knot complements
- Rasmussen
- 2003
(Show Context)
Citation Context ...rticular, the total rank of I \ .K/ is equal to the absolute value of the determinant of K. The group KHI.K/ closely resembles the “hat” version of Heegaard knot homology, bHF.K/, defined in [28] and =-=[31]-=-: one can perhaps think of KHI.K/ as the “instanton” counterpart of the “Heegaard” group bHF.K/. The present paper provides a spectral sequence from Khr.K/ to KHI.K/, but at the time4 of writing it i... |

101 | Holomorphic disks and knot invariants
- Ozsváth, Szabó
(Show Context)
Citation Context ...=2. In particular, the total rank of I \ .K/ is equal to the absolute value of the determinant of K. The group KHI.K/ closely resembles the “hat” version of Heegaard knot homology, bHF.K/, defined in =-=[28]-=- and [31]: one can perhaps think of KHI.K/ as the “instanton” counterpart of the “Heegaard” group bHF.K/. The present paper provides a spectral sequence from Khr.K/ to KHI.K/, but at the time4 of wri... |

94 |
An instanton-invariant for 3-manifolds
- Floer
(Show Context)
Citation Context ...p between the Khovanov cohomology of a knot, as defined in [16], and various homology theories defined using Yang-Mills instantons, of which the archetype is Floer’s instanton homology of 3-manifolds =-=[8]-=-. A consequence of this relationship is a proof that Khovanov cohomology detects the unknot. (For related results, see [10, 11, 12]). Theorem 1.1. A knot in S 3 is the unknot if and only if its reduce... |

87 | Khovanov homology and the slice genus
- Rasmussen
(Show Context)
Citation Context ...a diagram D in R2 . Let N be the number of crossings in the diagram. As in [16], we can consider the 2N possible smoothings of D, indexed by the points v of the cube f0; 1gN , with the conventions of =-=[16, 30]-=-, for example. This labeling of the smoothings is consistent with the convention illustrated in Figure 6. This gives 2N different unlinks Kv. For each v u in f0; 1gN , we have our standard cobordism S... |

70 |
The orientation of Yang-Mills moduli space and 4-manifolds topology
- Donaldson
- 1987
(Show Context)
Citation Context .... Proposition 2.12. The real line bundle det D of the family of operators D over B k .X; †; P/ is trivial. Proof. The proof follows that of the corresponding result in [21], which in turn is based on =-=[6]-=-. We must show that the determinant line is orientable along all closed loops in B .X; †; P/. The fundamental group of B .X; †; P/ is k k isomorphic to the group of components of GkC1.X; †; P/=f˙1g. T... |

68 | Gauge theory for embedded surfaces
- Kronheimer, Mrowka
- 1993
(Show Context)
Citation Context ...efinition of the invariant I \ .K/ by discussing instantons on 4-manifolds X with codimension-2 singularities along an embedded surface †. This is material that derives from the authors’ earlier work =-=[20]-=-, and it was developed further for arbitrary structure groups in [23]. In this paper we work only with the structure group SU.2/ or PSU.2/, but we extend the previous framework in two ways. First, in ... |

60 | Casson’s invariant and gauge theory
- Taubes
(Show Context)
Citation Context ...sversality properties for the both the set of critical points and the moduli spaces of trajectories for the formal gradient flow. Section 3.2 of [23] explains how to do this, following work of Taubes =-=[33]-=- and Donaldson; and the approach described there needs almost no modification in the present context. The basic function used in constructing perturbations is obtained as follows. Choose a lift of the... |

58 | On the Heegaard Floer homology of branched double-covers
- Ozsváth, Szabó
(Show Context)
Citation Context ...f pairs. Our main result concerning I \ .K/ is that it is related to reduced Khovanov cohomology by a spectral sequence. The model for this result is a closelyrelated theorem due to Ozsváth and Szabó =-=[29]-=- concerning the Heegaard Floer homology, with Z=2 coefficients, of a branched double cover of S 3 . There is a counterpart to the result of [29] in the context of Seiberg-Witten gauge theory, due to B... |

53 | An endomorphism of the Khovanov invariant - Lee - 2005 |

46 |
Floer’s work on instanton homology, knots and surgery. The Floer memorial volume
- Braam, Donaldson
- 1995
(Show Context)
Citation Context ...eterminant-1 gauge transformations. Our first task in this section is to outline how this is done. The issue appears (and is dealt with) already in Floer’s original proof of excision (as presented in =-=[4]-=-), but we need a more general framework. When enlarging the gauge group in this way, the standard approach to orienting moduli spaces breaks down, and an alternative method is needed. We turn to this ... |

39 |
T.: Monopoles and three-manifolds
- Kronheimer, Mrowka
- 2007
(Show Context)
Citation Context ...0/. In particular, each one-dimensional connected component ŒA� M M.ˇ1; ˇ0/1; being just a copy of R canonically oriented by the action of translations, determines an isomorphism ƒ.ˇ1/ ! ƒ.ˇ0/. As in =-=[23, 22]-=-, we denote by Zƒ.ˇ/ the infinite cyclic group whose two generators are the two elements of ƒ.ˇ/, and we denote by ŒA� M W Zƒ.ˇ1/ ! Zƒ.ˇ0/ the resulting isomorphism of groups. We now have everything w... |

34 | Monopoles and lens space surgeries - Kronheimer, Mrowka, et al. |

25 |
The index of elliptic operators over V -manifolds
- Kawasaki
- 1981
(Show Context)
Citation Context ... be the operator acting on the spaces D D d C A ˚ d A (7) LL 2 k . L XI g L P ˝ ƒ1 / ! LL 2 k 1 . L XI g L P ˝ .ƒC ˚ ƒ 0 //:22 Then in the orbifold setting D is a Fredholm operator. (See for example =-=[15]-=- and compare with [20, Proposition 4.17].) We now wish to define a moduli space of anti-self-dual connections as M.X; †; P/ D f A 2 C k j F C A D 0 g ı G kC1: Following [20], there is a Kuranishi mode... |

24 |
Classification of oriented sphere bundles over a 4-complex
- Dold, Whitney
- 1959
(Show Context)
Citation Context ...t exact sequence: .Z=2/ N � H 2 .X h � / � H 2 .Xn†/ ˚ H 2 .D/ �� H 2 .@ / �� �� H 2 .X/ �� H 2 .Xn†/ ˚ H 2 . / �� H 2 .@ / �� The lemma follows from an examination of the diagram. Let us recall from =-=[5]-=- that SO.3/ bundles P on a 4-dimensional simplicial complex Z can be classified as follows. First, P has a Stiefel-Whitney class w2.P /, which can take on any value in H 2 .ZI Z=2/. Second, the isomor... |

23 | On the Khovanov and knot Floer homologies of quasialternating links
- Manolescu, Ozsváth
- 2007
(Show Context)
Citation Context ...uasi-alternating) knots and links, it is known that the rank of the reduced Khovanov cohomology (over Q or over the field of 2 elements) is equal to the lower bound which the above corollary provides =-=[25, 27]-=-. Furthermore, that lower bound is simply the absolute value of the determinant in this case. We therefore deduce also: Corollary 1.6. When K quasi-alternating, the spectral sequence from Khr.K/ to I ... |

14 |
Knot polynomials and knot homologies. In Geometry and topology of manifolds
- Rasmussen
- 2005
(Show Context)
Citation Context ...uence from Khr.K/ to KHI.K/, but at the time4 of writing it is not known if there is a similar spectral sequence from Khr.K/ to bHF.K/ for classical knots. This was a question raised by Rasmussen in =-=[32]-=-, motivated by observed similarities between reduced Khovanov cohomology and Heegaard Floer homology. There are results in the direction of providing such a spectral sequence in [26], but the problem ... |

11 | Floer homology and surface decompositions
- Juhász
- 2008
(Show Context)
Citation Context ...gauge theory on a closed 3-manifold obtained from the knot complement. The construction of KHI.K/ in [24] was motivated by Juhász’s work on sutured3 manifolds in the setting of Heegaard Floer theory =-=[13, 14]-=-: in the context of sutured manifolds, KHI.K/ can be defined as the instanton Floer homology of the sutured 3-manifold obtained from the knot complement by placing two meridional sutures on the torus ... |

8 |
On the colored Jones polynomial, sutured Floer homology, and knot Floer homology, preprint
- Grigsby, Wehrli
- 2008
(Show Context)
Citation Context ...stantons, of which the archetype is Floer’s instanton homology of 3-manifolds [8]. A consequence of this relationship is a proof that Khovanov cohomology detects the unknot. (For related results, see =-=[10, 11, 12]-=-). Theorem 1.1. A knot in S 3 is the unknot if and only if its reduced Khovanov cohomology is Z. In [23], the authors construct a Floer homology for knots and links in 3manifolds using moduli spaces o... |

8 |
An obstruction to removing intersection points in immersed surfaces
- Kronheimer
- 1997
(Show Context)
Citation Context ...S/ is zero otherwise. Proof. The first point is that the map I ].S/ in this situation depends only on S as an abstract surface, not on its embedding in .0; 1/ R3 . This can be deduced from results of =-=[18]-=-, which show that the invariants of a closed pair .X; †/ defined using singular instantons depend on † only through its homotopy class. To apply the results of [18] to the present situation, we procee... |

8 | Knot homology groups from instantons - Kronheimer, Mrowka |

7 |
On the spectral sequence from Khovanov homology to Heegaard Floer homology
- Baldwin
(Show Context)
Citation Context ...rises from the112 spectral sequence is a topological invariant of K. More generally, one can ask whether the intermediate pages of the spectral sequence, as filtered groups, are invariants of K (see =-=[2]-=- for the similar question concerning the spectral sequence of Ozsváth and Szabó). A related question is whether the intermediate pages are functorial for knot cobordisms. Although the bigrading is abs... |

7 | A link surgery spectral sequence in monopole Floer homology, preprint
- Bloom
- 2009
(Show Context)
Citation Context ...erning the Heegaard Floer homology, with Z=2 coefficients, of a branched double cover of S 3 . There is a counterpart to the result of [29] in the context of Seiberg-Witten gauge theory, due to Bloom =-=[3]-=-. Proposition 1.2. With Z coefficients, there is a spectral sequence whose E2 term is the reduced Khovanov cohomology, Khr. K/, N of the mirror image knot K, N and which abuts to the reduced singular ... |

7 |
Instanton homology, surgery and knots, Geometry of low dimensional manifolds: 1
- Floer
- 1989
(Show Context)
Citation Context ...K/ has rank bigger than 1 for non-trivial knots. This will be done by relating I \ .K/ to a knot homology that was constructed from a different point of view (without singular instantons) by Floer in =-=[9]-=-. Floer’s knot homology was revisited by the authors in [24], where it appears as an invariant KHI.K/ of knots in S 3 . (There is a slight difference between KHI.K/ and Floer’s original version, in th... |

6 |
Holomorphic discs and sutured manifolds
- Juhász
(Show Context)
Citation Context ...gauge theory on a closed 3-manifold obtained from the knot complement. The construction of KHI.K/ in [24] was motivated by Juhász’s work on sutured3 manifolds in the setting of Heegaard Floer theory =-=[13, 14]-=-: in the context of sutured manifolds, KHI.K/ can be defined as the instanton Floer homology of the sutured 3-manifold obtained from the knot complement by placing two meridional sutures on the torus ... |

6 | An unoriented skein exact triangle for knot Floer homology
- Manolescu
- 2006
(Show Context)
Citation Context ...d by Rasmussen in [32], motivated by observed similarities between reduced Khovanov cohomology and Heegaard Floer homology. There are results in the direction of providing such a spectral sequence in =-=[26]-=-, but the problem remains open. 1.2 Outline Section 2 provides the framework for the definition of the invariant I \ .K/ by discussing instantons on 4-manifolds X with codimension-2 singularities alon... |

5 | Does Khovanov homology detect the unknot?, preprint
- Hedden, Watson
- 2008
(Show Context)
Citation Context ...stantons, of which the archetype is Floer’s instanton homology of 3-manifolds [8]. A consequence of this relationship is a proof that Khovanov cohomology detects the unknot. (For related results, see =-=[10, 11, 12]-=-). Theorem 1.1. A knot in S 3 is the unknot if and only if its reduced Khovanov cohomology is Z. In [23], the authors construct a Floer homology for knots and links in 3manifolds using moduli spaces o... |

3 |
Torsion classes and a universal constraint on Donaldson invariants for odd manifolds
- Akbulut, Mrowka, et al.
- 1995
(Show Context)
Citation Context ...roup of components, 0.K/, admits a surjective map 0.K/ ! H1.Xn†I Z/ whose kernel is either trivial or Z=2. The latter occurs precisely when w2.P / D w2.Xn†/ in H 2 .Xn†I Z=2/. Proof. This is standard =-=[1]-=-. A representative g for an element of 0.K/ is a section of G that is 1 on †. The corresponding element of H1.X n †I Z/ is represented by Qg 1 . 1/, where Qg ' g is a section transverse to 1. The kern... |

3 | Khovanov homology of the 2-cable detects the unknot. math.GT/0805.4418
- Hedden
- 2008
(Show Context)
Citation Context ...stantons, of which the archetype is Floer’s instanton homology of 3-manifolds [8]. A consequence of this relationship is a proof that Khovanov cohomology detects the unknot. (For related results, see =-=[10, 11, 12]-=-). Theorem 1.1. A knot in S 3 is the unknot if and only if its reduced Khovanov cohomology is Z. In [23], the authors construct a Floer homology for knots and links in 3manifolds using moduli spaces o... |

3 | Instanton Floer homology and the Alexander polynomial, preprint (manuscript
- Kronheimer, Mrowka
(Show Context)
Citation Context ...hen be reformulated as an isomorphism over Z between I \ .K/ and KHI.K/. Corollary 1.3 and Proposition 1.4 yield other lower bounds on the rank of the Khovanov cohomology. For example, it is shown in =-=[19]-=- that the Alexander polynomial of a knot can be obtained as the graded Euler characteristic for a certain decomposition of KHI.K/, so we can deduce: Corollary 1.5. The rank of the reduced Khovanov coh... |

3 |
sutures and excision
- Knots
- 2008
(Show Context)
Citation Context ...be done by relating I \ .K/ to a knot homology that was constructed from a different point of view (without singular instantons) by Floer in [9]. Floer’s knot homology was revisited by the authors in =-=[24]-=-, where it appears as an invariant KHI.K/ of knots in S 3 . (There is a slight difference between KHI.K/ and Floer’s original version, in that the latter leads to a group with twice the rank). It is d... |