Contributions to Khovanov Homology

Khovanov homology ist a new link invariant, discovered by M. Khovanov, and used by J. Rasmussen to give a combinatorial proof of the Milnor conjecture. In this thesis, we give examples of mutant links with different Khovanov homology. We prove that Khovanov's chain complex retracts to a subcomplex, whose generators are related to spanning trees of the Tait graph, and we exploit this result to investigate the structure of Khovanov homology for alternating knots. Further, we extend Rasmussen's invariant to links. Finally, we generalize Khovanov's categorifications of the colored Jones polynomial, and study conditions under which our categorifications are functorial with respect to colored framed link cobordisms. In this context, we develop a theory of Carter--Saito movie moves for framed link cobordisms.Comment: 77 pages; PhD thesis, Zurich, 200

Khovanov Homology and Conway Mutation

We present an easy example of mutant links with different Khovanov homology. The existence of such an example is important because it shows that Khovanov homology cannot be defined with a skein rule similar to the skein relation for the Jones polynomial.Comment: 9 pages, 5 figure

A spanning tree model for Khovanov homology

We use a spanning tree model to prove a result of E. S. Lee on the support of Khovanov homology of alternating knots.Comment: 13 pages, 3 figures; Footnote 1 was changed, a remark was rephrased as a theorem, a reference was added, typos were correcte

A remark on the topology of (n,n) Springer varieties

We prove a conjecture of Khovanov which identifies the topological space underlying the Springer variety of complete flags in C^2n stabilized by a fixed nilpotent operator with two Jordan blocks of size n.Comment: 8 pages, 1 figur