Beyond Möbius Bagels: Twisty Bagels

Rarely do Joe Carter and MAKE Magazine point me towards the same online curiosity . The intersection of topology and breakfast must have considerable ecumenical appeal. I’m going to turn Hart’s challenge around, however: now that you know how to create two interlocking bagel-halves by performing a twisty cut on a simple space, I’d like you to try to produce something topologically equivalent to two interlocking bagel-halves by performing a simple cut on a twisty space.

Here’s a hint:

Imagine that I have unrolled a Möbius strip. I’ll get something that looks like this:

It should be easy to see how we can recreate the original Möbius strip by gluing together the top and bottom edges (marked ‘A’) in such a way that the arrows align.

Now consider a line drawn around the circumference of the strip. On our unrolled strip, it will look like the dashed line ‘B’ below:

I will state but not prove that there is no way to continuously deform the line B to form a new line B’ which both also forms a complete loop wrapping around the strip AND does not intersect our original line B at any point. This should be intuitively quite obvious.

Now, what does this tell us about the space(s) that will result from cutting the Möbius strip along line B? Generalize what you’ve found to tori and you will have discovered the Twisty Bagel.

What have we learned? That sometimes the cut is relative to the space. Determining whether that is always the case would require clarification of our hazy concepts “cut” and “space”.