Indexed on: 21 Dec '17Published on: 21 Dec '17Published in: arXiv - Mathematics - Geometric Topology
There are no fewer than eight link homology theories which admit spectral sequences from Khovanov homology. Some of these theories are defined similarly to Khovanov homology, while others come from gauge theory (e.g. monopole Floer homology) and symplectic topology (e.g. Heegaard Floer homology) applied to branched double covers of links. Baldwin, Hedden, and Lobb call such gadgets *Khovanov-Floer theories* and abstractly characterize them with a few simple rules (arXiv:1509.04691). These rules are enough to prove that every Khovanov-Floer theory is Reidemeister-invariant and functorial, in the sense that link cobordisms induce maps of spectral sequences which are invariant under isotopies of the cobordism. Their theorem proves the value of the characterization and opens the possibility that the totality of such theories could be understood or even parametrized. But it crucially relies upon spectral sequence techniques. Therefore the functoriality they prove really is the functoriality of Khovanov-Floer theories per se, not the functoriality of (say) Heegaard Floer homology of branched double covers. In this paper, we define *strong Khovanov-Floer theories* using filtered chain complexes rather than spectral sequences. We prove Reidemeister-invariance and functoriality at the level of *chain homotopy classes of maps* rather than maps of spectral sequences. A priori, this is a stronger statement than Baldwin, Hedden, and Lobb's. Our results follow from the observation that generalized Khovanov-Floer theories factor through Bar-Natan's cobordism-theoretic framework for link homology. We prove that a strong Khovanov-Floer theory (satisfying one technical condition) induces a Khovanov-Floer theory.