Handle Cancellation in Flow Categories and the Khovanov Stable Homotopy Type

Abstract

This thesis extends classical handle cancellation occuring in Morse theory to framed flow categories. A particular framed flow category, the Khovanov flow category, was defined by Lipshitz-Sarkar in [LS14a] where they construct a Khovanov stable homotopy type. This stable homotopy tye induces a Steenrod square on Khovanov homology, and a result by Baues [Bau95] shows that this is enough to completely determine the Khovanov stable homotopy type of relatively simple links. This includes all links with up to 11 crossings, and [LS14b] provides a list of the stable homotopy types for all such links. The first knot for which these computations are non-trival is 8_, and the calculations for the Steenrod square of this knot can be simplified drastically using handle cancellation in framed flow categories. The thesis concludes by exhibiting this simplification.