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

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.