Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 15, Number 3 (2015), 1239-1302.
Framed graphs and the non-local ideal in the knot Floer cube of resolutions
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.
Algebr. Geom. Topol., Volume 15, Number 3 (2015), 1239-1302.
Received: 19 August 2013
Revised: 23 June 2014
Accepted: 27 August 2014
First available in Project Euclid: 16 November 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 57M27: Invariants of knots and 3-manifolds
Gilmore, Allison. Framed graphs and the non-local ideal in the knot Floer cube of resolutions. Algebr. Geom. Topol. 15 (2015), no. 3, 1239--1302. doi:10.2140/agt.2015.15.1239. https://projecteuclid.org/euclid.agt/1510840961