Khovanov homology and knot Floer homology for pointed links

TL;DR

This paper introduces a modified Khovanov homology for multi-basepoint links and a spectral sequence linking it to knot Floer homology, providing a framework to potentially prove a conjecture relating their ranks.

Contribution

It develops a new version of Khovanov homology for multi-basepoint links and constructs a spectral sequence to connect it with knot Floer homology, aiming to prove a key conjecture.

Findings

Constructed a modified Khovanov homology for links with multiple basepoints.

Introduced a spectral sequence converging to knot Floer homology.

Provided evidence supporting the conjecture over $\

Abstract

A well-known conjecture states that for any $l$-component link $L$ in $S^3$, the rank of the knot Floer homology of $L$ (over any field) is less than or equal to $2^{l-1}$ times the rank of the reduced Khovanov homology of $L$. In this paper, we describe a framework that might be used to prove this conjecture. We construct a modified version of Khovanov homology for links with multiple basepoints and show that it mimics the behavior of knot Floer homology. We also introduce a new spectral sequence converging to knot Floer homology whose $E_1$ page is conjecturally isomorphic to our new version of Khovanov homology; this would prove that the conjecture stated above holds over the field $\mathbb{Z}_2$.