A well-known conjecture states that for any l-component link L in S3, the rank of the knot Floer homology of L (over any field) is less than or equal to 2l-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 E1 page is conjecturally isomorphic to our new version of Khovanov homology; this would prove that the conjecture stated above holds over the field Z2.
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory
- Khovanov homology
- knot Floer homology