stable homotopy types of small cohomological width.

Plan of the paper {#subsec:plan}
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In Section \[sec:ffcs\], we give a condensed recap of the necessary definitions of a framed flow category. This is based on the formulation by Lipshitz and Sarkar [@MR3230817], and is intended as a quick-reference guide for the rest of the paper. In Section \[sec:handleslide\] we introduce an analogue for framed flow categories of the operation of *handle sliding* in a handlebody decomposition of a smooth manifold. In Section \[sec:whitney\] we similarly introduce an analogue of the *Whitney trick*.

Handle sliding, the extended Whitney trick, and handle cancellation (already introduced in [@JLSmorse]) are each moves that can performed on framed flow categories while preserving the associated stable homotopy type. We give an example of their application in Section \[sec:tref\]. Recently, Lipshitz and Sarkar [@MR3230817] have constructed a framed flow category from the input of a knot diagram. This gives a stable homotopy type whose cohomology recovers Khovanov cohomology (a knot invariant intimately connected with Floer cohomology). Our example is drawn from a particular quantum degree ($q = 21$) of the Khovanov stable homotopy type of the disjoint union of three right-handed trefoils.

Finally, in Section \[sec:smith\] we prove 