3
$\begingroup$

I was watching this video by Numberphile where a professor cuts a bagel into two interlocking pieces. Is this a torus knot or torus link?

I'm trying to interpret in terms of $(p,q)$-torus knots Torus knot (Wikipedia), but it seems like the cut goes once around the axis of rotational symmetry, and once around the circle in the torus, but I think I'm interpreting it wrong because since $\gcd(1,1)=1$, you shouldn't expect two components, even though that is what happens.

$\endgroup$

2 Answers 2

2
$\begingroup$

You might be familiar with how a torus can be thought of as identifying sides of a square. If not, look at this little gif from wikipedia: Identification square. A torus knot can be though of as a line with rational slope that passes through the square and when you get to an edge, you move to the opposite edge and continue on at the same slope.

So, a $(1,1) Torus knot is the diagonal of the square, which looks like this. A square with the diagonal drawn through it

But if you were to take a cylinder and stick a knife through it, you would cut it in two places. A cylinder with a line passing through and intersecting in two spots

If you were to follow the line as it cuts through the torus, or for us, the cylinder (we are cutting it before it would be bend around to make the usual torus) , you will see we get two independent lines that intersect the cylinder. One that starts at the bottom and one that starts at the top, which the second picture shows in red and black, respectively.

A cylinder that is sliced by a plane that twists through it. Similar to the previous picture, but now the two intersections are shown in red and black.

So, the corresponding identification square should have two $(1,1)$ torus knots, one in black and one in red.

Two (1,1) torus knots, one on the diagonal, the second shifted up half way from the first.

Hope this was helpful.

$\endgroup$
1
$\begingroup$

Every half donut is orientable, since it has two faces, hence it has two border components, and, as you suggest, they are both $(1,1)$ torus knots. The problem is that they are two torus knots, not one, hence you can get two connected components.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.