Khovanov Homotopy Calculations using Flow Category Calculus

Abstract

The Lipshitz–Sarkar stable homotopy link invariant defines Steenrod squares on the Khovanov cohomology of a link. Lipshitz–Sarkar constructed an algorithm for computing the first two Steenrod squares. We develop a new algorithm which implements the flow category simplification techniques previously defined by the authors and Dan Jones. We give a purely combinatorial approach to calculating the second Steenrod square and Bockstein homomorphisms in Khovanov cohomology, and flow categories in general. The new method has been implemented in a computer program by the third author and applied to large classes of knots and links. Several homotopy types not previously witnessed are observed, and more evidence is obtained that Khovanov stable homotopy types do not contain $\mathbb{C}{P}^2$ as a wedge summand. In fact, we are led by our calculations to formulate an even stronger conjecture in terms of $\mathbb{Z}/2$ summands of the cohomology.

Keywords:

• Steenrod square
• flow category
• Khovanov

2000 AMS SUBJECT CLASSIFICATION:

• 57M27

Notes

1 See [Baues 95, §11] for the definition of Chang space, and also for the definition of the various Baues-Hennes spaces.

Additional information

Funding

The authors were partially supported by the EPSRC Grant EP/K00591X/1. PO was supported by a CIRGET postdoctoral fellowship.