Deliberations in The Trial of the Punctured Torus

Can a torus (the surface of an inner tube or a doughnut) be turned inside-out? Is there one way to do it or more than one way? We will explore the merits and demerits of two possible ways to turn a torus inside-out presented by Mr. Jones and Dr. Situs in "The Trial of the Punctured Torus" which is Chapter 9 in Stephen Barr's fun book "Experiments in Topology".

If you are new to the book reading pages 1-9 in Chapter 1, pages 20-34 in Chapter 2, and all of Chapter 9 (pages 136-148) will provide sufficient background.

Here are the questions that will guide our exploration of the subject:
● Do you understand Dr. Situs' procedure for turning the punctured torus inside out? Can you explain it? Can you demonstrate it?
● Explain the resemblance (or lack thereof) between the statue of Laocoön and His Sons (see https://en.wikipedia.org/wiki/Laoco%C3%B6n_and_His_Sons) and the drawings in Figures 1-9 on pages 136-139.
● Do you agree with Mr. Jones that Dr. Situs cannot complete the inside-outing process beyond Figure 9 on page 139? Do you agree with Jones that Situs' torus is not really inside-out?
● Is Dr. Situs' allegedly inside-out torus homeomorphically equivalent to a torus? Or is it something else? What?
● What is the point of Mr. Jones' demonstration of turning a glove inside-out (see Figures 15-19 on pages 141-142)? If even one finger is not turned inside-out is the whole, therefore, not inverted? Is the partly inside-out glove homeomorphic to a right-handed glove?
● Do you agree with Mr. Jones that the two linked closed curves (the dotted and dashed lines in Figures 3-13 on pages 138-140) stay linked at the end of Dr. Situs' inversion of the punctured torus? Is Mr. Jones correct that "two closed curves that are linked cannot be unlinked by any topological [homeomorphic] deformation"? If true, does that prove that Dr. Situs' inversion is inadequate?
● In Figures 22-26 on page 144, Mr. Jones describes his procedure for inside-outing a torus. Is his cut of the torus an admissible homeomorphic deformation because he glues it back together with the same connectedness it had before he cut it? Is Mr. Jones' inversion valid? Is his approach better or worse than Dr. Situs'? Why?
● Do the linked dotted and dashed closed curves in Figures 22-26 on page 144 show that Mr. Jones' inversion is valid and Dr. Situs' is invalid? Is linkedness a topological invariant?
● Do both Dr. Situs' and Mr. Jones' inversions turn surface fur from the outside to the inside and vice versa as depicted in Figures 27-30 on page 146? Does that prove that both inversions are valid?
● Do both Dr. Situs' and Mr. Jones' inversions turn the grain of the surfaces from cylindrical to annular as in Figures 31-32 on page 147? Is this grain transformation a requirement for inside-outing?
● The deliberations in this trial consider inversions of a torus that 1) change the inner surface to the outer surface, 2) change the "grain" from cylindrical to annular, 3) preserve or break linked Jordan curves. Are any of these criteria requirements for inverting a torus? Which ones? Why?
● Do you consider Dr. Situs' or Mr. Jones' inversion to be valid? Which, if any, is invalid? Why?
● What are the topological requirements for inside-outing? How would you define topological inversion of surfaces?

This is the last topic in a series exploring Stephen Barr's fun book "Experiments in Topology".

"Experiments in Topology" is available from Dover (http://store.doverpublications.com/0486259331.html)

"Experiments in Topology" at Google Books (https://books.google.com/books?id=9TMx6ABV-98C)