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Quantum Topology

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Volume 7, Issue 1, 2016, pp. 107–183
DOI: 10.4171/QT/74

Published online: 2016-02-08

Invariance and the knot Floer cube of resolutions

Allison L. Gilmore[1]

(1) University of California Los Angeles, USA

This paper considers the invariance of knot Floer homology in a purely algebraic setting, without reference to Heegaard diagrams, holomorphic disks, or grid diagrams. We show that (a small modication of) Ozsváth and Szabó’s cube of resolutions for knot Floer homology, which is assigned to a braid presentation with a basepoint, is invariant under braid-like Reidemeister moves II and III and under conjugation. All moves are assumed to happen away from the basepoint. We also describe the behavior of the cube of resolutions under stabilization. The techniques echo those employed to prove the invariance of HOMFLY-PT homology by Khovanov and Rozansky, and are further evidence of a close relationship between the theories. The key idea is to prove categoried versions of certain equalities satisfied by the Murakami–Ohtsuki–Yamada state model for the HOMFLY-PT polynomial.

Keywords: Knot homologies, knot Floer homology, HOMFLY-PT homology, cube of resolutions, MOY relations

Gilmore Allison: Invariance and the knot Floer cube of resolutions. Quantum Topol. 7 (2016), 107-183. doi: 10.4171/QT/74