On the anti-diagonal filtration for the Heegaard Floer chain complex of a branched double-cover

TL;DR

This paper proves that the higher pages of a spectral sequence related to symplectic Khovanov cohomology and Heegaard Floer homology are invariants of knots, extending previous results about the E^{1}-page.

Contribution

It demonstrates that all pages of Seidel and Smith's spectral sequence are knot invariants, not just the E^{1}-page, advancing understanding of knot invariants in symplectic and Floer homology.

Findings

Higher pages of the spectral sequence are knot invariants.

Extends invariance from E^{1}-page to all pages.

Connects symplectic Khovanov cohomology with Heegaard Floer homology.

Abstract

Seidel and Smith introduced the graded fixed-point symplectic Khovanov cohomology group Kh_{symp,inv}(K) for a knot K inside S^{3}, as well as a spectral sequence converging to the Heegaard Floer homology-hat group for the connected sum of the double branched cover with a copy of S^{2}xS^{1}. The E^{1}-page of this spectral sequence is isomorphic to a factor of Kh_{symp,inv}(K). Seidel and Smith proved that Kh_{symp,inv} is a knot invariant. We show here that the higher pages of their spectral sequence are knot invariants also.