I wonder whether you mean that a `flow’ on the space of embedded spheres taking any sphere to the round sphere induces a flow on the space of unknotted circles taking any unknotted

circle to the round circle. I don’t see how would that work though.

But that sounds like a fun project.

]]>intersect planes containing the z-axis along simple closed curves so it seems spheres of 0-complexity can flow to the standard sphere by, say, using Grayson’s theorem to flow all these simple closed curves to standard circles. Dynnikov’s spheres and moves are combinatorially defined but I don’t see why they couldn’t be done smoothly to smooth spheres. One difficulty maybe is that he does these moves to spheres in a `general position’ with respect to the `foliation’ of R^3 by planes containing the z axis. ]]>