An untwisted cube of resolutions for knot Floer homology

Abstract

Ozsváth and Szabó gave a combinatorial description of knot Floer homology based on a cube of resolutions, which uses maps with twisted coefficients. We study the $t=1$ specialization of their construction. The associated spectral sequence converges to knot Floer homology, and we conjecture that its $E_1$ page is isomorphic to the HOMFLY-PT chain complex of Khovanov and Rozansky. At the level of each $E_1$ summand, this conjecture can be stated in terms of an isomorphism between certain Tor groups. As evidence for the conjecture, we prove that such an isomorphism exists in degree zero.