Categorifications of the colored Jones polynomial

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.Comment: 23 pages, latex, 16 eps figure

Matrix factorizations and link homology II

To a presentation of an oriented link as the closure of a braid we assign a complex of bigraded vector spaces. The Euler characteristic of this complex (and of its triply-graded cohomology groups) is the HOMFLYPT polynomial of the link. We show that the dimension of each cohomology group is a link invariant.Comment: 37 pages, 20 figures; version 2 corrects an inaccuracy in the proof of Proposition

Crossingless matchings and the cohomology of (n,n) Springer varieties

The sequence of rings $H^n, n\ge 0,$ introduced in math.QA/0103190, controls categorification of the quantum sl(2) invariant of tangles. We prove that the center of $H^n$ is isomorphic to the cohomology ring of the (n,n) Springer variety and show that the braid group action in the derived category of $H^n$-modules descends to the Springer action of the symmetric group.Comment: 20 pages, 7 figure

An invariant of tangle cobordisms

We prove that the construction of our previous paper math.QA/0103190 yields an invariant of tangle cobordisms.Comment: latex, 18 pages, 9 eps figure

Link homology and Frobenius extensions

We explain how rank two Frobenius extensions of commutative rings lead to link homology theories and discuss relations between these theories, Bar-Natan theories, equivariant cohomology and the Rasmussen invariant.Comment: 13 pages, 2 figure