On the functoriality of Khovanov-Floer theories

TL;DR

This paper introduces Khovanov-Floer theories, establishing their functoriality and showing how they induce spectral sequences that are link invariants and functorial under cobordisms, unifying several known spectral sequences.

Contribution

It formalizes the concept of Khovanov-Floer theories, proving their functoriality and applying this framework to known spectral sequences, providing new insights and potential invariants.

Findings

Spectral sequences from Khovanov-Floer theories are functorial and link invariants.

Several existing spectral sequences, including those from Heegaard Floer and instanton homology, are induced by Khovanov-Floer theories.

The framework offers new proofs of functoriality and defines potential new knot invariants.

Abstract

We introduce the notion of a Khovanov-Floer theory. Roughly, such a theory assigns a filtered chain complex over Z/2 to a link diagram such that (1) the E_2 page of the resulting spectral sequence is naturally isomorphic to the Khovanov homology of the link; (2) this filtered complex behaves nicely under planar isotopy, disjoint union, and 1-handle addition; and (3) the spectral sequence collapses at the E_2 page for any diagram of the unlink. We prove that a Khovanov-Floer theory naturally yields a functor from the link cobordism category to the category of spectral sequences. In particular, every page (after E_1) of the spectral sequence accompanying a Khovanov-Floer theory is a link invariant, and an oriented cobordism in R^3 \times [0,1] between links in R^3 induces a map between each page of their spectral sequences, invariant up to smooth isotopy of the cobordism rel boundary. We then show that the spectral sequences relating Khovanov homology to Heegaard Floer homology and singular instanton knot homology are induced by Khovanov-Floer theories and are therefore functorial in the manner described above, as has been conjectured for some time. We further show that Szabo's geometric spectral sequence comes from a Khovanov-Floer theory, and is thus functorial as well. In addition, we illustrate how our framework can be used to give another proof that Lee's spectral sequence is functorial and that Rasmussen's invariant is a knot invariant. Finally, we use this machinery to define some potentially new knot invariants.