Contributions to Khovanov Homology

Details Contributions to Khovanov Homology Contributions to Khovanov Homology

2008-10-05

arxiv.org/abs/0810.0778v1

Mathematics

Geometric Topology

77 pages; PhD thesis, Zurich, 2007

Scientific paper

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.

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