Geometry and Topology Seminar
Both knot Floer homology and HOMFLY-PT homology can be expressed in terms of an oriented cube of resolutions. Manolescu conjectured that the homologies of these two complexes with respect to their vertex maps are isomorphic, and that this isomorphism commutes with the induced edge maps. This would give a construction of the conjectured spectral sequence from HOMFLY-PT homology to HFK. We will prove that such an isomorphism exists by adding new differentials to the knot Floer complex which give an analog to the sl(n) differentials on HOMFLY-PT homology. Unfortunately, the isomorphism is not explicit, so commutativity with the edge maps remains open.