Knot Floer homology and Khovanov–Rozansky homology for singular links

Abstract

The (untwisted) oriented cube of resolutions for knot Floer homology assigns a complex $C_F\left(S\right)$ to a singular resolution $S$ of a knot $K$. Manolescu conjectured that when $S$ is in braid position, the homology $H_{\ast }\left(C_F\left(S\right)\right)$ is isomorphic to the HOMFLY-PT homology of $S$. Together with a naturality condition on the induced edge maps, this conjecture would prove the existence of a spectral sequence from HOMFLY-PT homology to knot Floer homology. Using a basepoint filtration on $C_F\left(S\right)$, a recursion formula for HOMFLY-PT homology and additional ${\mathfrak{s}\mathfrak{l}}_n$–like differentials on $C_F\left(S\right)$, we prove Manolescu’s conjecture. The naturality condition remains open.