Threading a two-holed torus which is hanging on a string.

Question

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.

Answer 1

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!

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

Answer 2

Hint

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