A Spectral Sequence from Khovanov Homology to Knot Floer Homology

Abstract

A well-known conjecture of Rasmussen states that for any knot K K in S 3 S^{3} , the rank of the reduced Khovanov homology of K K is greater than or equal to the rank of the reduced knot Floer homology of K K . This rank inequality is supposed to arise as the result of a spectral sequence from Khovanov homology to knot Floer homology. Using an oriented cube of resolutions construction for a homology theory related to knot Floer homology, we prove this conjecture.