Similar to the Floer homotopy type results Jonny cites, there are a few recent papers by Lipshitz-Sarkar Lipshitz-Sarkar on constructing a spectrum whose (singular) homology is Khovanov homology.
Besides giving an alternate construction of these various homologies, and the intrinsically interesting question of what spaces/spectra might underlie (or at least be related to) the Floer-theoretic constructions, I think there was some hope that other topological invariants of the spectra (e.g. generalized homology theories) would give new interesting invariants attached to, say, the underlying 3-manifold (in the Seiberg-Witten case). But I don't know whether anything along these lines ever panned out.
