Framed graphs and the non-local ideal in the knot Floer cube of resolutions

TL;DR

This paper introduces a homology theory for framed trivalent graphs that unifies aspects of knot Floer and HOMFLY-PT homologies, clarifying the role of the non-local ideal and its algebraic structure.

Contribution

It defines a new homology theory for framed graphs that bridges knot Floer and HOMFLY-PT invariants, and explains the non-local ideal via ideal quotients and Gr"obner basis techniques.

Findings

Homology theory for framed trivalent graphs is developed.

The non-local ideal is expressed as an ideal quotient.

Closing a braid strand corresponds to an ideal quotient operation.

Abstract

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.