This article addresses the two significant aspects of Ozsváth and Szabó’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 Gröbner basis argument that
connects the combinatorics of the non-local ideal to those of Buchberger’s
algorithm.
