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

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

Hint

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