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.

In principle, yes, perhaps Dynnikov’s ideas could be used to inspire an explicit potential function on the space of knots. But there are several rather large technical obstructions — Dynnikov’s constructions are diagrammatic so you would need to “smooth” it to the entire space and (this might be a reflection of my hazy recollection of what Dynnikov does) secondly, Dynnikov has a rather large decision procedure so it’s unclear how to make anything geometric out of something not so simple.

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. ]]>

It’s not clear how to use the fact that you’re on the unknot component to your advantage. Finding critical points of these energy functionals isn’t so easy. I’m probably going to give a related project to an undergrad — finding critical points of this functional, not on the unknot space, but on the space of a “big” satellite knot. There’s at least enough knowledge of the homotopy-type to get a sense for how to `cast the net’.

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