Sure it's honest. IF by that you mean: ----- Is it an isotopy* of the torus T^2 minus a disk D into R^3 that starts with a standard torus with a small round disk removed, and ends with the torus-with-a-hole occupying the same image in R^3 but with the generators** interchanged? ----- You can easily see that it's honest by letting it progress slowly (I have to start and stop it to do this; YMMV.) It's easy to follow either of the two types of holes** and see that it ends up as the other type. (OK, the teensy cheat is only that the removed disk ends up elsewhere from where it started, but a rotation about the z-axis will easily fix that.) —Dan ——————————————————————— * that is, a continuous family of homeomorphisms, or in technical terms, a map H: H: (T^2 - D)x[0,1] —> R^3 ** the conventional types of basis "holes" in a torus: a longitude (the long way around) and a meridian (the short way). -----Original Message-----
From: James Propp <jamespropp@gmail.com> Sent: Jul 14, 2016 1:27 PM To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] Everting a torus
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 _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun