In this question from mathematica.stackexchange it is asserted that the following two surfaces are homotopic. My basic question is - is this true?
It's clear to me that the two surfaces are homeomorphic, but homotopic seems more difficult to see, since the homotopy must disentangle the apparently knotted structure in the box. I don't actually doubt it, I just don't see it easily.
If they are homotopic, is it possible to exhibit the homotopy explicitly? If so, it might be possible to generate an animation illustrating the homotopy.
The first step of the homotopy will simply straighten out the left most tube: let the strand that loops around the middle tube pass back leftward through that middle tube, so that the leftmost tube then just has a trefoil knot tied in it; and then let the trefoil knot pass through itself to straighten out the left most tube so that it is completely vertical just like the other two tubes.
The second step of the homotopy is a (non-smooth) isotopy: simply round out the angled edges of the right picture, and then flatten it out somewhat; you get the left picture.