Link homology and categorification

We construct a bigraded cohomology theory of links whose Euler characteristic is the Jones polynomial.

This article consists of six lectures on the categorification of the Burau representation and on link homology groups which categorify the Jones and the HOMFLY-PT polynomial. The notes are based on the lecture course at the PCMI 2006 summer school in Park City, Utah.

This is a brief review of the categorification of the Jones polynomial and its significance and ramifications in geometry, algebra, and low-dimensional topology.

We present a short and unified representation-theoretical treatment of type A link invariants (that is, the HOMFLY-PT polynomials, the Jones polynomial, the Alexander polynomial and, more generally, the gl(m|n) quantum invariants) as link invariants with values in the quantized oriented Brauer category.

A braid-like isotopy for links in 3-space is an isotopy which uses only those Reidemeister moves which occur in isotopies of braids. We define a refined Jones polynomial and its corresponding Khovanov homology which are, in general, only invariant under braid-like isotopies.

In this thesis we work with Khovanov homology of links and its generalizations, as well as with the homology of graphs. Khovanov homology of links consists of graded chain complexes which are link invariants, up to chain homotopy, with graded Euler characteristic equal to the Jones polynomial of the link. Hence, it can be regarded as the "categorif\mbox{}ication" of the Jones polynomial. \indent We prove that the f\mbox{}irst homology group of positive braid knots is trivial....

The colored Jones polynomial of links has two natural normalizations: one in which the n-colored unknot evaluates to [n+1], the quantum dimension of the (n+1)-dimensional irreducible representation of quantum sl(2), and the other in which it evaluates to 1. For each normalization we construct a bigraded cohomology theory of links with the colored Jones polynomial as the Euler characteristic.

Khovanov homology offers a nontrivial generalization of Jones polynomial of links in R^3 (and of Kauffman bracket skein module of some 3-manifolds). In this chapter (Chapter X) we define Khovanov homology of links in R^3 and generalize the construction into links in an I-bundle over a surface. We use Viro's approach to construction of Khovanov homology and utilize the fact that one works with unoriented diagrams (unoriented framed links) in which case there is a simple descri...

There is a generalization of Heegaard-Floer theory from ${\mathfrak{gl}}_{1|1}$ to other Lie (super)algebras $^L{\mathfrak{g}}$. The corresponding category of A-branes is solvable explicitly and categorifies quantum $U_q(^L{\mathfrak{g}})$ link invariants. The theory was discovered in \cite{A1,A2}, using homological mirror symmetry. It has novel features, including equivariance and, if $^L{\mathfrak{g}} \neq {\mathfrak{gl}}_{1|1}$, coefficients in categories. In this paper, w...

The definition of the Jones polynomial in the 80's gave rise to a large family of so-called quantum link invariants, based on quantum groups. These quantum invariants are all controlled by the same two-variable invariant (the HOMFLY-PT polynomial), which also specializes to the older Alexander polynomial. Building upon quantum Schur--Weyl duality and variants of this phenomenon, I will explain an algebraic setup that allows for global definitions of these quantum polynomials,...