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

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

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

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

Categorification of the colored Jones polynomial and Rasmussen invariant of links

We define a family of formal Khovanov brackets of a colored link depending on two parameters. The isomorphism classes of these brackets are invariants of framed colored links. The Bar-Natan functors applied to these brackets produce Khovanov and Lee homology theories categorifying the colored Jones polynomial. Further, we study conditions under which framed colored link cobordisms induce chain transformations between our formal brackets. We conjecture that, for special choice of parameters, Khovanov and Lee homology theories of colored links are functorial (up to sign). Finally, we extend the Rasmussen invariant to links and give examples, where this invariant is a stronger obstruction to sliceness than the multivariable Levine-Tristram signature.Comment: 26 pages with figures. Minor revisions. We weakened the statement of Lemma 6.1, whose original proof was incomplet

On the naturality of the spectral sequence from Khovanov homology to Heegaard Floer homology

Ozsvath and Szabo have established an algebraic relationship, in the form of a spectral sequence, between the reduced Khovanov homology of (the mirror of) a link L in S^3 and the Heegaard Floer homology of its double-branched cover. This relationship has since been recast by the authors as a specific instance of a broader connection between Khovanov- and Heegaard Floer-type homology theories, using a version of Heegaard Floer homology for sutured manifolds developed by Juhasz. In the present work we prove the naturality of the spectral sequence under certain elementary TQFT operations, using a generalization of Juhasz's surface decomposition theorem valid for decomposing surfaces geometrically disjoint from an imbedded framed link.Comment: 36 pages, 13 figure

On Gradings in Khovanov homology and sutured Floer homology

We discuss generalizations of Ozsvath-Szabo's spectral sequence relating Khovanov homology and Heegaard Floer homology, focusing attention on an explicit relationship between natural Z (resp., 1/2 Z) gradings appearing in the two theories. These two gradings have simple representation-theoretic (resp., geometric) interpretations, which we also review.Comment: 17 pages, 5 figures, to be submitted to Proceedings of Jaco's 70th Birthday Conference, 201