Khovanov's homology for tangles and cobordisms
@article{BarNatan2004KhovanovsHF, title={Khovanov's homology for tangles and cobordisms}, author={Dror Bar-Natan}, journal={Geometry \& Topology}, year={2004}, volume={9}, pages={1443-1499} }
We give a fresh introduction to the Khovanov Homology theory for knots and links, with special emphasis on its extension to tangles, cobordisms and 2-knots. By staying within a world of topological pictures a little longer than in other articles on the subject, the required extension becomes essentially tautological. And then a simple application of an appropriate functor (a "TQFT") to our pictures takes them to the familiar realm of complexes of (graded) vector spaces and ordinary homological…
Figures from this paper
458 Citations
An sl(2) tangle homology and seamed cobordisms
- Mathematics
- 2007
We construct a bigraded (co)homology theory which depends on a parameter a, and whose graded Euler characteristic is the quantum sl(2) link invariant. We follow Bar-Natan's approach to tangles on one…
Twisted skein homology
- Mathematics
- 2012
We apply the techniques of totally twisted Khovanov homology to Asaeda, Przytycki, and Sikora's construction of Khovanov type homologies for links and tangles in I-bundles over (orientable) surfaces.…
The finiteness result for Khovanov homology and localization in monoidal categories
- Mathematics
- 2008
In the previous paper we constructed the local system of Khovanov complexes on the Vassiliev space of knots and extended it to the singular locus. In this paper we introduce the definition of the…
A Khovanov type link homology with geometric interpretation
- Mathematics
- 2016
We study a Khovanov type homology close to the original Khovanov homology theory from Frobenius system. The homology is an invariant for oriented links up to isotopy by applying a tautological…
sl(2) tangle homology with a parameter and singular cobordisms
- Mathematics
- 2008
We construct a bigraded cohomology theory which depends on one parameter a, and whose graded Euler characteristic is the quantum sl.2/ link invariant. We follow Bar-Natan’s approach to tangles on one…
On a triply graded Khovanov homology
- Mathematics
- 2015
Cobordisms are naturally bigraded and we show that this grading extends to Khovanov homology, making it a triply graded theory. Although the new grading does not make the homology a stronger…
Fixing the functoriality of Khovanov homology: a simple approach
- MathematicsJournal of Knot Theory and Its Ramifications
- 2021
Khovanov homology is functorial up to sign with respect to link cobordisms. The sign indeterminacy has been fixed by several authors, by extending the original theory both conceptually and…
The Two Formulations of Khovanov’s Tangle Invariant
- Mathematics
- 2018
In [2], Khovanov introduces a homology theory for links, which he calls a ‘categorification’ of the Jones polynomial. To an oriented link projection he assigns a chain complex of graded abelian…
Khovanov homology and strong inversions
- Mathematics
- 2021
Abstract. There is a one-to-one correspondence between strong inversions on knots in the three-sphere and a special class of four-ended tangles. We compute the reduced Khovanov homology of such…
gl(2) foams and the Khovanov homotopy type
- Mathematics
- 2021
The Blanchet link homology theory is an oriented model of Khovanov homology, functorial over the integers with respect to link cobordisms. We formulate a stable homotopy refinement of the Blanchet…
References
SHOWING 1-10 OF 34 REFERENCES
An invariant of link cobordisms from Khovanov homology.
- Mathematics
- 2004
In (10), Mikhail Khovanov constructed a homology theory for oriented links, whose graded Euler characteristic is the Jones polynomial. He also explained how every link cobordism between two links…
Remarks on definition of Khovanov homology
- Mathematics
- 2002
Mikhail Khovanov in math.QA/9908171 defined, for a diagram of an oriented classical link, a collection of groups numerated by pairs of integers. These groups were constructed as homology groups of…
An invariant of link cobordisms from Khovanov's homology theory
- Mathematics
- 2002
Mikhail Khovanov constructed (math.QA/9908171) a homology theory of oriented links, which has the Jones polynomial as its graded Euler characteristic. He also explained how every link cobordism…
Remarks on Definition of Khovanov Homology
- Mathematics
- 2002
Mikhail Khovanov defined, for a diagram of an oriented classical link, a collection of groups numerated by pairs of integers. These groups were constructed as homology groups of certain chain…
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…
On Khovanov’s categorification of the Jones polynomial
- Mathematics
- 2002
The working mathematician fears complicated words but loves pictures and diagrams. We thus give a no-fancy-anything picture rich glimpse into Khovanov's novel construction of "the categorification of…
Torsion of the Khovanov homology
- Mathematics
- 2004
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…
An invariant of tangle cobordisms
- Mathematics
- 2002
We construct a new invariant of tangle cobordisms. The invariant of a tangle is a complex of bimodules over certain rings, well-defined up to chain homotopy equivalence. The invariant of a tangle…
Categorification of the Kauffman bracket skein module of I-bundles over surfaces
- Mathematics
- 2004
Khovanov defined graded homology groups for links LR 3 and showed that their polynomial Euler characteristic is the Jones polyno- mial of L. Khovanov's construction does not extend in a…
A functor-valued invariant of tangles
- Mathematics
- 2002
We construct a family of rings. To a plane diagram of a tangle we associate a complex of bimodules over these rings. Chain homotopy equivalence class of this complex is an invariant of the tangle. On…