A knot Floer stable homotopy type

TL;DR

This paper constructs a filtered spectrum from grid diagrams of knots, conjecturing it as a knot invariant, and introduces a combinatorial approach to defining the associated homotopy type without holomorphic geometry.

Contribution

It introduces a new combinatorial method to define the knot Floer homotopy type directly from grid diagrams, bypassing holomorphic techniques.

Findings

Constructed a filtered spectrum from grid diagrams of knots.

Conjectured the homotopy type as a knot invariant.

Developed a combinatorial framework for moduli spaces.

Abstract

Given a grid diagram for a knot or link K in $S^3$, we construct a filtered spectrum whose homology is the knot Floer homology of K. We conjecture that the filtered homotopy type of the spectrum is an invariant of K. Our construction does not use holomorphic geometry, but rather builds on the combinatorial definition of grid homology. We inductively define models for the moduli spaces of pseudo-holomorphic strips and disk bubbles, and patch them together into a framed flow category. The inductive step relies on the vanishing of an obstruction class that takes values in a complex of positive domains with partitions.