Link invariants related to knot Floer homology and Khovanov homology
Link invariants related to knot Floer homology and Khovanov homology
Despite the differences in their constructions, knot Floer homology and Khovanov-Rozansky homology seem to have a great deal in common. I will introduce a family of invariants HFK_n on the knot Floer side which are the knot Floer analogs of sl_n homology. These invariants don't readily allow a cube of resolutions construction, but in the n=2 case I will give an algebraically constructed complex which is expected to be quasi-isomorphic to HFK_2. This complex does decompose as an oriented cube of resolutions, and we will show that the E_2 page of the associated spectral sequence is isomorphic to Khovanov homology. Since reduced HFK_2 is isomorphic to delta-graded HFK, this gives a possible construction of the spectral sequence from Khovanov homology to knot Floer homology. Joint with Akram Alishahi.