I have a question about a specific immersion of the Klein bottle in $\mathbb{R}^{3}$ that seems curious in different aspects to me. First, it is obtained by a mechanical movement of a geometrical circle that is easy to define analitically. Second, each circle $L$ from this family is meeting exactly two others and $L$ is linked with the circles that are 'close' to it, while is unlinked with the 'distant' ones. This model reminds the 'figure 8' immersion of the Klein bottle (Klein bagel), but is quite different, as the Klein bagel is obtained by mechanical movement of a 2:1 Lissajous curve similar to figure 8 (see wiki).

The idea is to take a 'small' circle $l$ and a 'big' circle $L$ so that the center of $L$ lies on $l$, both in one and the same plane. Then $L$ starts moving so that its center is describing $l$, and simultaneously $L$ is turning around $l$ (in fact - around the tangent line to $l$ at the current center of $L$). It suffices to adjust the speeds so that when the center of $L$ makes a full round of $l$, circle $L$ achieves a half turn around $l$. Then $L$ describes an immersed Klein bottle.

Now, my questions about this model are:

1) Is everything OK with this immersion (at least topologically) since it is not quite easy to visualise the result.

2) How exactly the self-intersection set looks like? I suppose it consists of one or two closed curves. (By the way, are there any general results saying what the self-intersection set of an immersed Klein bottle should be?)

3) Are there any references about similar 'mechanical' models of surfaces in $\mathbb{R}^{3}$?

Note that if everything is alright with the model, we can take a small knot $l$ instead of a circle and to make $L$ turning $n+1/2$ times around $l$, in such a way getting many different immersions of the Klein bottle.

Anyway, let me write the analytical description of this model (up to my fault), although I am not sure it helps understanding the things geometrically.

Here $l:x^{2}+y^{2}=1,\ z=0$, $L:x^{2}+(y-1)^{2}=r^{2},\ z=0;\ r>2$. The immersion is then given by

$X=r\cos u\cos2t-\sin2t\cos t(1+r\sin u)$

$Y=r\cos u\sin2t+\cos2t\cos t(1+r\sin u)$

$Z=\sin t(1+r\sin u)$,

where $t\in\lbrack0,\pi],\ u\in\lbrack0,2\pi]$.

Note that for $r\leq2$ the same example should be valid as well; especially for $r<1$ it is easier to visualise, but the case $r>2$ seems more enigmatic.



1 Answer 1


Not an answer; just an aid to visualization. If I've interpreted your equations correctly, here are views of the surfaces for three values of $r$:

  • $\begingroup$ Thank you very much, Joseph, this task is /almost/ impossible to me at present! Seems very twisty! $\endgroup$
    – t22
    Apr 30, 2011 at 16:13

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