On spectral sequences from Khovanov homology

Abstract

There are a number of homological knot invariants, each satisfying an unoriented skein exact sequence, which can be realized as the limit page of a spectral sequence starting at a version of the Khovanov chain complex. Compositions of elementary 1-handle movie moves induce a morphism of spectral sequences. These morphisms remain unexploited in the literature, perhaps because there is still an open question concerning the naturality of maps induced by general movies.In this paper we focus on the spectral sequences due to Kronheimer-Mrowka from Khovanov homology to instanton knot Floer homology, and on that due to Ozsváth-Szabó to the Heegaard-Floer homology of the branched double cover. For example, we use the 1-handle morphisms to give new information about the filtrations on the instanton knot Floer homology of the (4, 5)-torus knot, determining these up to an ambiguity in a pair of degrees; to determine the Ozsváth-Szabó spectral sequence for an infinite class of prime knots; and to show that higher differentials of both the Kronheimer-Mrowka and the Ozsváth-Szabó spectral sequences necessarily lower the delta grading for all pretzel knots.

On the functoriality of Khovanov–Floer theories

“…in Theorem 1.3. The third author and Zentner [26] recently used the idea that diagrammatic 1-handle additions induce morphisms of spectral sequences to compute the singular instanton and Heegaard Floer spectral sequences for a variety of knots, even without the assumption (proved in this paper) that the morphism associated to a movie for a cobordism is independent of the movie. The functoriality established here should allow us to extend these sorts of calculations to a wider array of knots.…”

The pillowcase and traceless representations of knot groups II: a Lagrangian–Floer theory in the pillowcase

“…As explained in [18, Proposition 3.1] (see Proposition 6.1 above), any representation of a, b, c, d | ba = cd taking a, b, c, d to traceless elements is conjugate to one given by (25) a We have shown that to each (γ, θ) ∈ P there exists a unique ρ ∈ R δ (S 2 × I, {a, b, c, d} × I), given by (25), (26), and (27). Moreover the restriction…”

“…Figure 16 shows the 4-punctured 2-sphere with the four based meridian generators a, b, c, d based at a point s. An additional curve e is also indicated. We have shown that to each (γ, θ) ∈ P there exists a unique ρ ∈ R δ (S 2 × I, {a, b, c, d} × I), given by ( 25), (26), and (27). Moreover the restriction…”

“…This is due in large part to the fact that computations are extremely scarce. Initially, the only route for computation was through the aforementioned spectral sequence, but, aside from the instances where it collapses for simple reasons at Khovanov homology, little headway has been made in this direction (though see [24,27] for more sophisticated computations using the spectral sequence).…”

On the Uniqueness of Sarkar-Seed-Szabo Construction

“…For a link homology theory pCpLq, d L q, it is natural for the deformation ǫ :" ǫ L to depend on the link L in such a manner that the chain complex pCpLq, d L `ǫL q remains invariant under the Reidemeister moves. Such constructions give rise to spectral sequences (see [10], [1], [7], [8]). Since there are many deformations of Khovanov homology, one can ask whether there are relations among them.…”