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

Allison Gilmore

Algebraic & Geometric Topology 15 (2015) 1239–1302

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.

framed graph, knot Floer homology, HOMFLY-PT homology, ideal quotient
Mathematical Subject Classification 2010
Primary: 57M27
Received: 19 August 2013
Revised: 23 June 2014
Accepted: 27 August 2014
Published: 19 June 2015
Allison Gilmore
Department of Mathematics
University of California, Los Angeles
520 Portola Plaza
Los Angeles, CA 90095