Compare the easy-to-follow choreography at https://www.youtube.com/watch?v=S4ddRPvwcZI (due to Greg McShane, I'm guessing) with the much subtler choreography at https://www.youtube.com/watch?v=jA86M6fdm_Q (due to Arnaud Cheritat, I'm guessing). Is the former eversion honest? Note that a circular patch has been cut away from the torus to make the action easier to follow; this permits unintended cheating. After all, you could make a video of a sphere-with-a-hole-in-it being turned inside out, and it would look convincing, unless you think hard about what happens to the boundary of the patch that bounds the hole. The same goes for a recent gif of Simon Pampena's (which I can view on Twitter but can't find on the web). Jim Propp