Geodesic
Framed graphs and the non-local ideal in the knot Floer cube of resolutions
Gilmore, Allison1
1Department of Mathematics, University of California, Los Angeles, 520 Portola Plaza, Los Angeles, CA 90095, USA
Algebraic and Geometric Topology, Tome 15 (2015) no. 3, pp. 1239-1302
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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.

DOI : 10.2140/agt.2015.15.1239
Classification : 57M27
Keywords: framed graph, knot Floer homology, HOMFLY-PT homology, ideal quotient

Gilmore, Allison  1

1 Department of Mathematics, University of California, Los Angeles, 520 Portola Plaza, Los Angeles, CA 90095, USA 
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Gilmore, Allison. Framed graphs and the non-local ideal in the knot Floer cube of resolutions. Algebraic and Geometric Topology, Tome 15 (2015) no. 3, pp. 1239-1302. doi: 10.2140/agt.2015.15.1239

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