Invariance and the knot Floer cube of resolutions

Th is paper considers the invariance of knot Floer homology in a purely algebraic setting, without reference to Heegaard diagrams, holomorphic disks, or grid diagrams. We show that (a small modi cation of) Ozsváth and Szabó’s cube of resolutions for knot Floer homology, which is assigned to a braid presentation with a basepoint, is invariant under braid-like Reidemeister moves II and III and under conjugation. All moves are assumed to happen away from the basepoint. We also describe the behavior of the cube of resolutions under stabilization. The techniques echo those employed to prove the invariance of HOMFLY-PT homology by Khovanov and Rozansky, and are further evidence of a close relationship between the theories. The key idea is to prove categori ed versions of certain equalities satis fied by the Murakami–Ohtsuki–Yamada state model for the HOMFLY-PT polynomial.