Functoriality Results for Khovanov’s Link Homology

My research at UCSD has focused on advancing our understanding of Khovanov’s homology theory for links and link cobordisms, first introduced in [10]. His theory is a “categorification” of the Jones polynomial V (L), so described because it associates to any link L a complex Kh(L) of bigraded modules whose graded Euler characteristic is V (L). Not only is the homotopy type of Kh(L) a link invariant, but it contains geometric information about cobordisms into and out of L (ie, knotted surfaces in 4-space) that appears nowhere in V (L) itself. Khovanov homology thus presents an exciting opportunity to enrich our knowledge of the powerful yet poorly-understood Jones polynomial and related quantum invariants. My results so far specifically address the nature of this cobordism information.