A knot Floer stable homotopy type

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.

123 pages; changed the marking to be avoided from X to O, so that our construction gives a filtered spectrum; added Theorem 1.1 and its proof (Section 15), changed the models for internal framings (Section 7.3), improved the discussion of signs (Sections 4.2 and 12), added new examples (Section 16); many other smaller changes throughout

References in corpus (7)

• Floer homology and knot complements
• Heegaard Floer homology and alternating knots
• Holomorphic disks and link invariants
• A Khovanov stable homotopy type
• Parametrized Ring-Spectra and the Nearby Lagrangian Conjecture
• Khovanov homotopy type, Burnside category, and products
• Unfolded Seiberg-Witten Floer spectra, I: Definition and invariance