Invariance and the knot Floer cube of resolutions
Abstract
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 modification of) Ozsv\'ath and Szab\'o'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 categorified versions of certain equalities satisfied by the Murakami-Ohtsuki-Yamada state model for the HOMFLY-PT polynomial.
Keywords
Cite
@article{arxiv.1007.2609,
title = {Invariance and the knot Floer cube of resolutions},
author = {Allison Gilmore},
journal= {arXiv preprint arXiv:1007.2609},
year = {2014}
}
Comments
68 pages, 31 figures; v3 to appear in Quantum Topology; v2 substantially revised from v1 including weakened statement of main result due to error in v1's Lemma 5 (conjugation / moving the basepoint). Other changes: substantial elaboration of argument for Reidemeister 3 invariance, different descriptions of behavior under stabilization and adding/removing layers of marked points