String through 1 hole 3-torus

Question

Okay So I had stayed up way too late thinking about this problem and I typed my question wrong. The question is: How do I deform a 3 dimensional 1 hole torus to go around a line? https://i.stack.imgur.com/Xb0yn.jpg

How do I deform a torus to go through the hole of a 3-torus? I have searched and searched but I cant wrap my brain around it (hehe, get it?).
I can understand a 2-hole and deforming a two hole torus with a line through it and getting both holes to go around the line, but not the how to deform it for the first line.

Thanks in advance.

that makes me remember http://upload.wikimedia.org/wikipedia/commons/b/ba/Inside-out_torus_%28animated%2C_small%29.gif — janmarqz, Oct 22 '14 at 16:53
1 hole 3 dimensional torus... and also I do not believe we are allowed to have it be a punctured torus — Ryan Taus, Oct 22 '14 at 20:50
If I understand you, a 3-dimensional 1-hole torus is usually called a solid torus. However, I don't know what you mean by deforming it to go around a line. Are you thinking of it as embedded in 3D? Where is the line in relation to the solid torus initially? — Cheerful Parsnip, Oct 22 '14 at 22:16
Are you thinking about something like this? http://math.stackexchange.com/questions/107782/continuously-deform-2-torus-with-a-line-through-one-hole-to-make-it-go-through-b — Cheerful Parsnip, Oct 22 '14 at 22:18
Almost but it doesnt start in a hole, and the torus has only one..I will draw a picture right now. — Ryan Taus, Oct 22 '14 at 22:31

score 1 · Answer 1 · answered Oct 22 '14 at 23:07

1

Okay, now that I know what the problem is, I can tell you it's impossible. There is no way to link the torus with the line by continuous deformation. The concept of linking number is relevant here. All the curves on the original torus have zero linking number with the line, but in the linked picture, the core curve of the torus has linking number $\pm 1$ with the line depending on how you orient things. (I'm assuming here the line goes to infinity, which is a case where we can define linking number.)

answered Oct 22 '14 at 23:07

Cheerful Parsnip

27,278

Can it be done by puncturing or any way at all? My professor is insisting it is possible, but it could be that he is messing with us. And thanks for the help! – Ryan Taus Oct 22 '14 at 23:12
No, puncturing it doesn't help. The same argument applies. – Cheerful Parsnip Oct 22 '14 at 23:14

String through 1 hole 3-torus

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