A knot Floer stable homotopy type

Abstract

Given a grid diagram for a knot or link K in S 3 , we construct a spectrum whose homology is the knot Floer homology of K. We conjecture that the 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. Contents1. Introduction 2. Background 3. The complex of positive domains 4. The complex of positive domains with partitions 5. n -manifolds 6. Stratified spaces 7. Local models 8. Moduli spaces 9. The stratification 10. Embeddings and framings 11. The embedded framed cobordism group 12. Constructing the moduli spaces 13. Embedding and framing the permutohedra 14. The Cohen-Jones-Segal construction References

Involutions, knots, and Floer K-theory

“…These are closely related to the recent work by Baraglia and Hekmati [8], where they established an equivariant Seiberg-Witten Floer stable homotopy type for finite groups, and an equivariant Seiberg-Witten Floer cohomology, and defined knot invariants in a similar way. (Note also a recent combinatorial construction of knot Floer homotopy type by Manolescu and Sarkar [59].) However, even restricting our attention to involutions and the case of spin 3-manifolds, there are significant differences between the equivariant Floer homotopy type by Baraglia-Hekmati [8] and our Floer homotopy type for involutions.…”