Khovanov Homology - Related Theories

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 sl_n 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 sl_n 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.