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The problem concerns a two-holed torus and an infinite length of string that passes through one of the holes. The object is to use continuous transformations on either the 2-hold torus or the infinite piece of string so that the string threads the two holes of the torus, i.e., it enters one hole and leaves through the other. This is a problem posed in one of the video lectures of N J Wildberger on youtube algebraic topology lectures.

After a lot of head scratching, I am at a loss.

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    $\begingroup$ Try pinching the tips of your thumbs and forefingers together at a single point. Then slide your right forefinger up along your left forefinger, and your right thumb up along your left thumb. $\endgroup$ – Chris Culter Feb 9 '19 at 0:01
  • $\begingroup$ I think I've got it. String isn't the only thing that can go through a hole. $\endgroup$ – timtfj Feb 9 '19 at 0:35
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You can do a continuous transformation of a genus-$2$ surface so that it has obvious three-fold rotational symmetry. Once it is in this form, it's hard to say whether the string passes through one or two holes, or even how many holes a genus-$2$ surface is meant to have!

A genus-2 surface with a string running through it

David Richeson made a claymation video illustrating the transformation: https://www.youtube.com/watch?v=S5fPwE7GQOA

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  • $\begingroup$ +1 That might be the best drawing of that surface I’ve ever seen. Mine never do it justice. $\endgroup$ – Santana Afton Feb 10 '19 at 2:06
  • $\begingroup$ Very clear. Thank you. $\endgroup$ – Iconoclast Feb 10 '19 at 13:55
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Hint

Here’s a before-and-after picture with a blue loop on the surface for some perspective:

enter image description here

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  • $\begingroup$ You're basically poking the whole thing through the hole with the blue loop so the "inside" and "outside" of the loop change places? Or rather, poking the initially threaded end through tte initially unthreaded hole? $\endgroup$ – timtfj Feb 9 '19 at 0:32
  • $\begingroup$ @timtfj Exactly! $\endgroup$ – Santana Afton Feb 9 '19 at 3:39
  • $\begingroup$ Yes indeed. Need more hint. $\endgroup$ – Iconoclast Feb 9 '19 at 9:40

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