A Steenrod Square for Link Floer Homology

TL;DR

This work advances practical Steenrod-square computations in grid-based link Floer homology by developing a framed 1-flow-category approach that requires only lower-dimensional moduli data from the Manolescu–Sarkar construction. It introduces the complex of domains with partitions $ extit{CDP}_*$ and its finite subcomplexes, proves the existence of a frame assignment $f$ (up to coboundaries) via obstruction theory, and provides an algorithm to compute $Sq^2$ for all grid versions by building a framed 1-flow category in the Lobb–Orson–Schütz sense. The paper terms out explicit constructions, including sign and frame data, and demonstrates the method on a concrete $2 imes2$ unknot grid, revealing how different frame choices affect the resulting $Sq^2$ and clarifying which choices arise from a genuine framed flow category. Overall, it bridges the gap between grid-moduli handling and computable Steenrod operations, enabling systematic calculations in the stable-homotopy refinement of link Floer homology.

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-Schütz 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.

Figures (27)

• Figure 1: A $5 \times 5$ grid diagram for the trefoil, along with a generator of the grid chain complex drawn with •. Note that the generator is independent of the $X$ and $O$ markings.
• Figure 2: The composition $M_{1/\sqrt{2}(e_1 + e_2)}M_{e_1}$ equals the rotation from $e_1$ to $e_2$. Here vertex 1 is the origin throughout.
• Figure 3: Type 2.0a, in the case where $N_1 = 2, N_2 = 1$. Here $N_j$ denotes a cluster of $N$ type $j$ bubbles, and $\mathbb{R}_{+(j)}$ denotes the $j^{th}$$\mathbb{R}_+$ coordinate.
• Figure 4: Type 2.0b, in the case where $N_1 = 1, N_2 = 2$, with all conventions as before.
• Figure 5: Type 2.1, in the case where $N = 1$, with all conventions as before.

Theorems & Definitions (54)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 3.1
• Definition 3.3
• Definition 3.3