  This article addresses the two significant aspects of Ozsv\'ath and Szab\'o's
knot Floer cube of resolutions that differentiate it from Khovanov and
Rozansky's HOMFLY-PT chain complex: (1) the use of twisted coefficients and (2)
the appearance of a mysterious non-local ideal. Our goal is to facilitate
progress on Rasmussen's conjecture that a spectral sequence relates the two
knot homologies. We replace the language of twisted coefficients with the more
quantum topological language of framings on trivalent graphs. We define a
homology theory for framed trivalent graphs with boundary that -- for a
particular non-blackboard framing -- specializes to the homology of singular
knots underlying the knot Floer cube of resolutions. For blackboard framed
graphs, our theory conjecturally recovers the graph homology underlying the
HOMFLY-PT chain complex. We explain the appearance of the non-local ideal by
expressing it as an ideal quotient of an ideal that appears in both the
HOMFLY-PT and knot Floer cubes of resolutions. This result is a corollary of
our main theorem, which is that closing a strand in a braid graph corresponds
to taking an ideal quotient of its non-local ideal. The proof is a Gr\"obner
basis argument that connects the combinatorics of the non-local ideal to those
of Buchberger's Algorithm.
