An untwisted cube of resolutions for knot Floer homology

TL;DR

This paper investigates a simplified version of a cube of resolutions for knot Floer homology, proposing a conjecture linking it to the HOMFLY-PT chain complex and providing partial evidence through degree zero isomorphisms.

Contribution

It studies the t=1 specialization of Ozsvath and Szabo's cube of resolutions and conjectures an isomorphism with the HOMFLY-PT chain complex, offering initial evidence.

Findings

Spectral sequence converges to knot Floer homology.

Conjecture that E_1 page is isomorphic to HOMFLY-PT chain complex.

Proved isomorphism exists in degree zero.

Abstract

Ozsvath and Szabo 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.