# A categorification of the Jones polynomial

@article{Khovanov1999ACO, title={A categorification of the Jones polynomial}, author={Mikhail Khovanov}, journal={Duke Mathematical Journal}, year={1999}, volume={101}, pages={359-426} }

Author(s): Khovanov, Mikhail | Abstract: We construct a bigraded cohomology theory of links whose Euler characteristic is the Jones polynomial.

## 429 Citations

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Given a graph, we construct homology groups whose Euler characteristic is the Penrose polynomial of the graph, evaluated at an integer. This work is motivated by Khovanov's work on the…

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We prove that the bigraded, colored Khovanov–Rozansky type A link and tangle invariants are functorial with respect to link and tangle cobordisms.

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Abstract This short note resolves the most important part of the PS braid conjecture while introducing the first known examples of knots and links with odd torsion of order 9, 27, 81, and 25 in their…

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We offer an alternative construction of Roberts' totally twisted Khovanov homology and prove that it agrees with delta-graded reduced characteristic-2 Khovanov homology.

### THE KHOVANOV HOMOLOGY OF A FAMILY OF THREE-COLUMN PRETZEL LINKS

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We give recursive relations to compute the rational Khovanov bigraded homology groups of three infinite families of non-quasi-alternating 3-column pretzel links. Based on this computations, we obtain…

### REIDEMEISTER INVARIANCE FOR KHOVANOV HOMOLOGY

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- 2004

Mikhail Khovanov constructed cohomology groups whose graded Euler characteristic is Jones polynomial VL of invariant for links L. In other words, for a diagram D of L, Khovanov composed bigraded…

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### Algebraic and topological perspectives on the Khovanov homology

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We investigate the Khovanov homology, introduced in [4], of an embedded link. A detailed computation for the trefoil is provided, along with two diffirent proofs of invariance under Reidemeister…

## References

SHOWING 1-10 OF 19 REFERENCES

### An instanton-invariant for 3-manifolds

- Mathematics
- 1988

To an oriented closed 3-dimensional manifoldM withH1(M, ℤ)=0, we assign a ℤ8-graded homology groupI*(M) whose Euler characteristic is twice Casson's invariant. The definition uses a construction on…

### Four‐dimensional topological quantum field theory, Hopf categories, and the canonical bases

- Mathematics
- 1994

A new combinatorial method of constructing four‐dimensional topological quantum field theories is proposed. The method uses a new type of algebraic structure called a Hopf category. The construction…

### Introduction to Quantum Groups

- Mathematics
- 1998

We give an elementary introduction to the theory of algebraic and topological quantum groups (in the spirit of S. L. Woronowicz). In particular, we recall the basic facts from Hopf (*-) algebra…

### Casson's invariant for oriented homology 3-spheres : an exposition

- Mathematics
- 1990

In the spring of 1985, A. Casson announced an interesting invariant of homology 3-spheres via constructions on representation spaces. This invariant generalizes the Rohlin invariant and gives…

### A categorification of the Temperley-Lieb algebra and Schur quotients of $ U({\frak sl}_2) $ via projective and Zuckerman functors

- Mathematics
- 1999

Author(s): Bernstein, Joseph; Frenkel, Igor; Khovanov, Mikhail | Abstract: We identify the Grothendieck group of certain direct sum of singular blocks of the highest weight category for sl(n) with…

### Introduction to Quantum Groups

- Mathematics
- 1993

THE DRINFELD JIMBO ALGERBRA U.- The Algebra f.- Weyl Group, Root Datum.- The Algebra U.- The Quasi--Matrix.- The Symmetries of an Integrable U-Module.- Complete Reducibility Theorems.- Higher Order…

### A polynomial invariant for knots via von Neumann algebras

- Mathematics
- 1985

Thus, the trivial link with n components is represented by the pair (l ,n), and the unknot is represented by (si$2 * * • s n i , n) for any n, where si, $2, • • • > sn_i are the usual generators for…

### Quantum SU(2)-invariants dominate Casson's SU(2)-invariant

- MathematicsMathematical Proceedings of the Cambridge Philosophical Society
- 1994

In 1988, E. Witten proposed an array of invariants (quantum G-invariants) for a 3-manifold associated with a compact simple Lie group G based on the quantum field theory [22]. N. Reshetikhin and V.…

### 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…