Khovanov homotopy calculations using flow category calculus

TL;DR

This paper introduces a new combinatorial algorithm for computing Steenrod squares in Khovanov homology using flow category calculus, enabling analysis of complex knots and providing evidence for conjectures about stable homotopy types.

Contribution

The authors develop a purely combinatorial method for calculating the second Steenrod square and Bockstein homomorphisms, improving computational techniques in Khovanov homotopy theory.

Findings

Implemented in software and applied to large classes of knots and links.

Discovered new homotopy types not previously observed.

Provided evidence supporting the conjecture that certain summands are absent in Khovanov stable homotopy types.

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.