A Steenrod Square for Link Floer Homology

TL;DR

This paper constructs explicit framings for moduli spaces in link Floer homology, enabling the computation of the second Steenrod square via a new algorithm.

Contribution

It provides explicit framings for moduli spaces in link Floer homology and introduces an algorithm to compute the second Steenrod square.

Findings

Constructed explicit framings for moduli spaces in link Floer homology.

Developed an algorithm to compute the second Steenrod square for grid homology.

Enabled new computations in link Floer homology using the constructed framings.

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. We explicitly give the framings of the lower-dimensional moduli spaces of the Manolescu-Sarkar construction as well as the more general moduli spaces corresponding to the full grid. Though in the latter case the stable homotopy type is not known, the explicit framings are enough to construct a framed 1-flow category, a construction by Lobb-Orson-Sch\"utz which contains enough information to find the second Steenrod square. Finally, we find an algorithm for computing the second Steenrod square for all versions of grid homology coming from the full grid.