Khovanov-Rozansky Complexes in the Knot Floer Cube of Resolutions

Abstract

The (untwisted) oriented cube of resolutions for knot Floer homology assigns a complex C_{F} (S) to a singular resolution S of a knot K. Manolescu conjectured that when S is in braid position, the homology H(C_{F}(S)) is isomorphic to the HOMFLY-PT homology of S. Together with a naturality condition on the induced edge maps, this conjecture would prove the spectral sequence from HOMFLY-PT homology to knot Floer homology. Using a basepoint filtration on C_{F}(S), a recursion formula for HOMFLY-PT homology, and additional sln-like differentials on C_{F}(S), we prove this conjecture. Since the isomorphism is not explicitly defined, the naturality of the induced edge maps remains open.