I wanted these because I wanted the loop space of any manifold to be a smooth space of some sort, etc. One standard approach to this issue uses infinite-dimensional manifolds called Frechet manifolds. But it’s a bit of a pain to make the space of smooth maps between smooth manifolds into a Frechet manifold, and even more of a pain to make the space of smooth maps between Frechet manifolds into a Frechet manifold (basically you can’t). None of this is difficult if we use a better notion of “smooth space”. There are several notions that will work, and we study a few, but my favorite is diffeological spaces.

]]>(Note that I’m posting this privately at least in part because I know that he still does regular searches for me online, and I don’t want him to see this post.)

]]>Can we assume that “changing colours whenever it passes over or under it on a different colour” means the following: When inner color A passes under or over outer color B ≠ A, inner color A always changes to the third color C where A ≠ C ≠ B ?

]]>Look at the left trefoil-pair for example. Start from the bottom right internal arc, and walk around the knot. Green under blue turns red. Red under red stays red. Red over red stays red. Red under green turns blue. Blue over green turns red. Red under clue turns green, and we’re back where we started… ]]>

I’m sorry, I REALLY don’t understand how the picture with the parallel knots is described by the above “rule” you describe.

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