Higher Differentials on Khovanov Homology

Abstract

In this thesis, we study the structure and geometric content of Khovanov homology using higher differentials. We study the Szabo geometric spectral sequence and conjecture that it agrees with the spectral sequence from Khovanov homology to the Heegaard Floer homology of the double-branched cover of a knot. We define a twisted variant of the geometric spectral sequence, connect it to Baldwin-Ozsvath-Szabo homology, and outline a strategy towards the above conjecture. We construct a new spectral sequence that begins at the Khovanov homology of a link and converges to the Khovanov homology of the disjoint union of its components. The page at which the spectral sequence collapses gives a lower bound on the splitting number of the link, the minimum number of times its components must be passed through one another in order to completely separate them. In addition, we build on work of Kronheimer-Mrowka and Hedden-Ni to show that Khovanov homology detects the unlink.