• Corpus ID: 14771903

Link homology and categorification

  title={Link homology and categorification},
  author={Mikhail Khovanov},
This is a short survey of algebro-combinatorial link homology theories which have the Jones polynomial and other link polynomials as their Euler characteristics. 

Khovanov Homology Theories and Their Applications

This is an expository paper discussing various versions of Khovanov homology theories, interrelations between them, their properties, and their applications to other areas of knot theory and

Algebraic and topological perspectives on the Khovanov homology

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

Knot homology via derived categories of coherent sheaves I, sl(2) case

Using derived categories of equivariant coherent sheaves, we construct a categorification of the tangle calculus associated to sl(2) and its standard representation. Our construction is related to

Schur-Weyl Dualities and Link Homologies

In this note we describe a representation theoretic approach to functorial functor valued knot invariants with the focus on (categorified) Schur-Weyl dualities. Applications include categorified

The Alexander polynomial as an intersection of two cycles in a symmetric power

We consider a braid β which acts on a punctured plane. Then we construct a local system on this plane and find a homology cycle D in its symmetric power, such that D ⋅ β(D) coincides with the

Khovanov homology of the $2$-cable detects the unknot

Inspired by recent work of Grigsby and Werhli, we use the deep geometric content of Ozsvath and Szabo's Floer homology theory to provide a short proof that the Khovanov homology of the 2-cable

Knot homology via derived categories of coherent sheaves II, $\mathfrak{sl}_{m}$ case

Using derived categories of equivariant coherent sheaves we construct a knot homology theory which categorifies the quantum $\mathfrak{sl}_{m}$ knot polynomial. Our knot homology naturally satisfies

Derived equivalences and finite dimensional algebras

We discuss the homological algebra of representation theory of finite dimensional algebras and finite groups. We present various methods for the construction and the study of equivalences of derived

Introduction to 3-Manifolds

Perspectives on manifolds Surfaces 3-manifolds Knots and links in 3-manifolds Triangulated 3-manifolds Heegaard splittings Further topics General position Morse functions Bibliography Index


It is conjectured that the Khovanov homology of a knot is invariant under mutation. In this paper, we review the spanning tree complex for Khovanov homology, and reformulate this conjecture using a

sl(3) link homology

Author(s): Khovanov, Mikhail | Abstract: We define a bigraded homology theory whose Euler characteristic is the quantum sl(3) link invariant.

Link homology and Frobenius extensions

Author(s): Khovanov, Mikhail | Abstract: We explain how rank two Frobenius extensions of commutative rings lead to link homology theories and discuss relations between these theories, Bar-Natan

Khovanov-Rozansky homology of two-bridge knots and links

We compute the reduced version of Khovanov and Rozansky's sl(N) homology for two-bridge knots and links. The answer is expressed in terms of the HOMFLY polynomial and signature.


In this article, we prove the conjecture of Bar-Natan, Garoufalidis, and Khovanov’s on the support of the Khovanov’s invariants for alternating knots.

Khovanov homology and the slice genus

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

Matrix factorizations and link homology II

For each positive integer n the HOMFLY polynomial of links specializes to a one-variable polynomial that can be recovered from the representation theory of quantum sl(n). For each such n we build a

Matrix factorizations and Kauffman homology

The topological string interpretation of homological knot invariants has led to several insights into the structure of the theory in the case of sl(N). We study possible extensions of the matrix

Braids, transversal links and the Khovanov-Rozansky Theory

We establish some inequalities for the Khovanov-Rozansky cohomologies of braids. These give new upper bounds of the self-linking numbers of transversal links in standard contact S 3 which are sharper

Braids, Transversal Knots and the Khovanov-rozansky Theory

We establish some inequalities about the Khovanov-Rozansky cohomologies of braids. These give new upper bounds of the self-linking numbers of transversal knots in standard contact S which is sharper