# Fixing the functoriality of Khovanov homology

@article{Clark2009FixingTF, title={Fixing the functoriality of Khovanov homology}, author={David A. Clark and Scott Morrison and Kevin Walker}, journal={Geometry \& Topology}, year={2009}, volume={13}, pages={1499-1582} }

We describe a modification of Khovanov homology [13], in the spirit of Bar-Natan [2], which makes the theory properly functorial with respect to link cobordisms. This requires introducing “disorientations” in the category of smoothings and abstract cobordisms between them used in Bar-Natan’s definition. Disorientations have “seams” separating oppositely oriented regions, coming with a preferred normal direction. The seams satisfy certain relations (just as the underlying cobordisms satisfy…

## Figures from this paper

## 112 Citations

A 2-category of chronological cobordisms and odd Khovanov homology

- Mathematics
- 2013

We create a framework for odd Khovanov homology in the spirit of Bar-Natan's construction for the ordinary Khovanov homology. Namely, we express the cube of resolutions of a link diagram as a diagram…

A cobordism realizing crossing change on sl2 tangle homology and a categorified Vassiliev skein relation

- MathematicsTopology and its Applications
- 2021

Categorification and applications in topology and representation theory

- Mathematics
- 2013

This thesis splits into two major parts. The connection between the two parts is the notion of "categorification" which we shortly explain/recall in the introduction.
In the first part of this…

Khovanov homology and categorification of skein modules

- MathematicsQuantum Topology
- 2018

For every oriented surface of finite type, we construct a functorial Khovanov homology for links in a thickening of the surface, which takes values in a categorification of the corresponding gl(2)…

FUNCTORIALITY RESULTS FOR KHOVANOV’S LINK HOMOLOGY

- Mathematics
- 2007

My research at UCSD has focused on advancing our understanding of Khovanov’s homology theory for links and link cobordisms, first introduced in [10]. His theory is a “categorification” of the Jones…

A cobordism realizing crossing change on $\mathfrak{sl}_2$ tangle homology and a categorified Vassiliev skein relation

- Mathematics
- 2020

In this paper, we discuss degree 0 crossing change on Khovanov homology in terms of cobordisms. Namely, using Bar-Natan's formalism of Khovanov homology, we introduce a sum of cobordisms that yields…

Tensor product algebras, Grassmannians and Khovanov homology

- Mathematics
- 2013

We discuss a new perspective on Khovanov homology, using categorifications of tensor products. While in many ways more technically demanding than Khovanov's approach (and its extension by Bar-Natan),…

Khovanov homology and cobordisms between split links

- MathematicsJournal of Topology
- 2022

In this paper, we study the (in)sensitivity of the Khovanov functor to four-dimensional linking of surfaces. We prove that if $L$ and $L'$ are split links, and $C$ is a cobordism between $L$ and $L'$…

Khovanov Homology, Lee Homology and a Rasmussen Invariant for Virtual Knots

- Mathematics
- 2014

The paper contains an essentially self-contained treatment of Khovanov homology, Khovanov-Lee homology as well as the Rasmussen invariant for virtual knots and virtual knot cobordisms which directly…

## References

SHOWING 1-10 OF 25 REFERENCES

Khovanov-Rozansky homology via a canopolis formalism

- Mathematics
- 2007

In this paper, we describe a canopolis (ie categorified planar algebra) formalism for Khovanov and Rozansky’s link homology theory. We show how this allows us to organize simplifications in the…

The universal Khovanov link homology theory

- Mathematics
- 2006

We determine the algebraic structure underlying the geometric complex associated to a link in Bar-Natan’s geometric formalism of Khovanov’s link homology theory (nD 2). We find an isomorphism of…

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…

Khovanov's homology for tangles and cobordisms

- Mathematics
- 2004

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…

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…

On the Algebraic Structure of Bar-Natan's Universal Geometric Complex and the Geometric Structure of Khovanov Link Homology Theories

- Mathematics
- 2006

We determine the exact algebraic structure underlying the geometric complex associated to a link in Bar-Natan’s geometric formalism of Khovanov’s link homology theory (n = 2). We find a complex…

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…

REIDEMEISTER MOVES FOR SURFACE ISOTOPIES AND THEIR INTERPRETATION AS MOVES TO MOVIES

- Mathematics
- 1993

A movie description of a surface embedded in 4-space is a sequence of knot and link diagrams obtained from a projection of the surface to 3-space by taking 2-dimensional cross sections perpendicular…

A Combinatorial Description of Knotted Surfaces and Their Isotopies

- Mathematics
- 1997

Abstract We discuss the diagrammatic theory of knot isotopies in dimension 4. We project a knotted surface to a three-dimensional space and arrange the surface to have generic singularities upon…

FAST KHOVANOV HOMOLOGY COMPUTATIONS

- Mathematics
- 2006

We introduce a local algorithm for Khovanov homology computations — that is, we explain how it is possible to "cancel" terms in the Khovanov complex associated with a ("local") tangle, hence…