Knot Floer Homology and Khovanov-Rozansky Homology for Singular Links

TL;DR

This paper proves a conjecture relating knot Floer homology and HOMFLY-PT homology for singular links, establishing a spectral sequence connecting these two important knot invariants.

Contribution

The authors prove Manolescu's conjecture that the homology of the knot Floer complex for singular links in braid position matches HOMFLY-PT homology, confirming a key spectral sequence.

Findings

Homology of the complex $C_{F}(S)$ is isomorphic to HOMFLY-PT homology for braid position.

Established a recursion formula for HOMFLY-PT homology using a basepoint filtration.

Introduced $sl_{n}$-like differentials to prove the conjecture.

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 $sl_{n}$-like differentials on $C_{F}(S)$, we prove this conjecture.