Related Theories
One of the most interesting aspects of Khovanov's homology is that its exact sequences are formally similar to those arising in the Floer homology of 3-manifolds. Moreover, it has been used to reprove a result only demonstrated using gauge theory and its cousins: Jacob Rasmussen's new proof of a theorem of Kronheimer and Mrowka, formerly known as the Milnor conjecture (see below). Conjecturally, there is a spectral sequence relating Khovanov homology with the knot Floer homology of Peter Ozsváth and Zoltán Szabó (Dunfield et al. 2005). Another spectral sequence (Ozsváth-Szabó 2005) relates a variant of Khovanov homology with the Heegard Floer homology of the branched double cover along a knot. A third (Bloom 2009) converges to a variant of the monopole Floer homology of the branched double cover.
Khovanov homology is related to the representation theory of the Lie algebra sl2. Mikhail Khovanov and Lev Rozansky have since defined cohomology theories associated to sln for all n. In 2003, Catharina Stroppel extended Khovanov homology to an invariant of tangles (a categorified version of Reshetikhin-Turaev invariants) which also generalizes to sln for all n. Paul Seidel and Ivan Smith have constructed a singly graded knot homology theory using Lagrangian intersection Floer homology, which they conjecture to be isomorphic to a singly-graded version of Khovanov homology. Ciprian Manolescu has since simplified their construction and shown how to recover the Jones polynomial from the chain complex underlying his version of the Seidel-Smith invariant.
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