Obstruction Complexes in Grid Homology

TL;DR

This paper introduces the obstruction chain complex for full grid diagrams in link Floer homology, computes its homology, and discusses implications for sign assignments, advancing the understanding of grid homology structures.

Contribution

It defines the obstruction chain complex for full grid diagrams and computes its homology, extending previous work that only considered diagrams with a square removed.

Findings

Computed the homology of the obstruction chain complex for full grid diagrams.

Established results on the existence of sign assignments in grid homology.

Extended the obstruction complex framework to more general grid diagrams.

Abstract

Recently, Manolescu-Sarkar constructed a stable homotopy type for link Floer homology, which uses grid homology and accounts for all domains that do not pass through a specific square. In doing so, they produced an obstruction chain complex of the grid diagram with that square removed. We define the obstruction chain complex of the full grid, without the square removed, and compute its homology. Though this homology is too complicated to immediately extend the Manolescu-Sarkar construction, we give results about the existence of sign assignments in grid homology.