On the Spectral Sequence from Khovanov Homology to Heegaard Floer Homology

Abstract

Ozsvath and Szabo [“On the Heegaard Floer homology of branched double-covers.” Advances in Mathematics 194, no. 1 (2005): 1–33.] show that there is a spectral sequence the E 2 term of which is and which converges to . We prove that the E k term of this spectral sequence is an invariant of the link L for all k≥2. If L is a transverse link in (S 3 ,ξ std ), then we show that Plamenevskaya’s transverse invariant ψ(L) gives rise to a transverse invariant, ψ k (L), in the E k term for each k≥2. We use this fact to compute each term in the spectral sequences associated to the torus knots T(3,4) and T(3,5).