Abstract:
We show that the reduced sl(n) homology defined by Khovanov and Rozansky is invariant under component-preserving positive mutation when n is odd.

Abstract:
We investigate the Khovanov-Rozansky invariant of a certain tangle and its compositions. Surprisingly the complexes we encounter reduce to ones that are very simple. Furthermore, we discuss a "local" algorithm for computing Khovanov-Rozansky homology and compare our results with those for the "foam" version of sl_3-homology.

Abstract:
We study the relationship between the HOMFLY and sl(N) knot homologies introduced by Khovanov and Rozansky. For each N>0, we show there is a spectral sequence which starts at the HOMFLY homology and converges to the sl(N) homology. As an application, we determine the KR-homology of knots with 9 crossings or fewer.

Abstract:
We determine the cycle packing number of a directed graph using elementary projective algebraic geometry. Our idea is rooted in the Khovanov-Rozansky theory. In fact, using the Khovanov-Rozansky homology of a graph, we also obtain algebraic methods of detecting directed and undirected cycles containing a particular vertex or edge.

Abstract:
We introduce an sl(n) homology theory for knots and links in the thickened annulus. To do so, we first give a fresh perspective on sutured annular Khovanov homology, showing that its definition follows naturally from trace decategorifications of enhanced sl(2) foams and categorified quantum gl(m), via classical skew Howe duality. This framework then extends to give our annular sl(n) link homology theory, which we call sutured annular Khovanov-Rozansky homology. We show that the sl(n) sutured annular Khovanov-Rozansky homology of an annular link carries an action of the Lie algebra sl(n), which in the n=2 case recovers a result of Grigsby-Licata-Wehrli.

Abstract:
In analogy with a recursive formula for the HOMFLY-PT polynomial of links given by Jaeger, we give a recursive formula for the graph polynomial introduced by Kauffman and Vogel. We show how this formula extends to the Khovanov-Rozansky graph homology.

Abstract:
We give a detailed proof of the fact that for any Coxeter group the Euler characteristic of the corresponding Khovanov-Rozansky homology provides a Markov trace.

Abstract:
We conjecture a relation between the sl(N) knot homology, recently introduced by Khovanov and Rozansky, and the spectrum of BPS states captured by open topological strings. This conjecture leads to new regularities among the sl(N) knot homology groups and suggests that they can be interpreted directly in topological string theory. We use this approach in various examples to predict the sl(N) knot homology groups for all values of N. We verify that our predictions pass some non-trivial checks.

Abstract:
The untwisted cube of resolutions for knot Floer homology assigns a chain complex $C(S)$ to each singular resolution $S$ of a knot $K$. It was conjectured by Manolescu that the homology of this complex is isomorphic to the HOMFLY-PT homology of $S$ defined by Khovanov and Rozansky. We show that, like HOMFLY-PT homology, the homology of $C(S)$ has a family of spectral sequences $E_{k}(n)$ for $n \ge 2$ converging to the $sl_{n}$ homology of $S$.

Abstract:
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.