Invariance and the knot Floer cube of resolutions

TL;DR

This paper establishes the algebraic invariance of a knot Floer homology cube of resolutions under certain Reidemeister moves and conjugation, highlighting connections with HOMFLY-PT homology without relying on geometric tools.

Contribution

It introduces an algebraic approach to prove invariance of the knot Floer cube of resolutions, paralleling methods used for HOMFLY-PT homology, and analyzes its behavior under stabilization.

Findings

Cube of resolutions is invariant under braid-like Reidemeister moves II and III and conjugation.

The approach is purely algebraic, avoiding geometric or analytic methods.

Demonstrates a close relationship between knot Floer homology and HOMFLY-PT homology.

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.