# 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}
}```
• M. Khovanov
• Published 30 August 1999
• Mathematics
• Duke Mathematical Journal
Author(s): Khovanov, Mikhail | Abstract: We construct a bigraded cohomology theory of links whose Euler characteristic is the Jones polynomial.
717 Citations
An elementary construction of Khovanov-Rozansky type link homology
In this article, we give an elementary construction of homological invariants of links presented by braid closures. The Euler characteristic of this complex is equal to quantum polynomial invariantExpand
A quantum categorification of the Alexander polynomial
• Mathematics
• 2019
Using a modified foam evaluation, we give a categorification of the Alexander polynomial of a knot. We also give a purely algebraic version of this knot homology which makes it appear as the infiniteExpand
This is a short survey of algebro-combinatorial link homology theories which have the Jones polynomial and other link polynomials as their Euler characteristics.
A CATEGORIFICATION FOR THE PENROSE POLYNOMIAL
• Mathematics
• 2011
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 theExpand
New proofs of certain finite filling results via Khovanov homology
We give a Khovanov homology proof that hyperbolic twist knots do not admit non-trivial Dehn surgeries with finite fundamental group.
• Mathematics
• Proceedings of the London Mathematical Society
• 2018
We prove that the bigraded colored Khovanov-Rozansky type A link and tangle invariants are functorial with respect to link and tangle cobordisms.
Khovanov's invariant for closed surfaces
We compute the Khovanov-Jacobsson number of an embedded torus in R^4. The answer is always 2, regardless of the embedding.
J ul 2 01 3 A geometric spectral sequence in Khovanov homology
The aim of this paper is to introduce and study a geometric spectral sequence on Z2 Khovanov homology. AMS Classification Keywords
A geometric spectral sequence in Khovanov homology
The aim of this paper is to introduce and study a geometric spectral sequence in Khovanov homology. The construction was motivated by a similar spectral sequence from Khovanov homology to HeegaardExpand
q,t-Catalan numbers and knot homology
We propose an algebraic model of the conjectural triply graded homology of Gukov, Dunfield and Rasmussen for some torus knots. It turns out to be related to the q,t-Catalan numbers of Garsia andExpand

#### References

SHOWING 1-10 OF 14 REFERENCES
An instanton-invariant for 3-manifolds
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 onExpand
Four‐dimensional topological quantum field theory, Hopf categories, and the canonical bases
• Mathematics, Physics
• 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 constructionExpand
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 givesExpand
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) withExpand
On the computational complexity of the Jones and Tutte polynomials
• Mathematics
• 1990
We show that determining the Jones polynomial of an alternating link is #P-hard. This is a special case of a wide range of results on the general intractability of the evaluation of the TutteExpand
Introduction to Quantum Groups
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 OrderExpand
State Models and the Jones Polynomial
IN THIS PAPER I construct a state model for the (original) Jones polynomial [5]. (In [6] a state model was constructed for the Conway polynomial.) As we shall see, this model for the Jones polynomialExpand
A polynomial invariant for knots via von Neumann algebras
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 forExpand
Quantum SU(2)-invariants dominate Casson's SU(2)-invariant
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.Expand
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 perpendicularExpand