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what is parameterization e.g., (x(u,v),y(u,v),z(u,v)), of the Trefoil knot surface in R³ whose cross section of the surface can be circular, or in general elliptic?


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You mean the Seifeit surface? –  Kerry Mar 17 '12 at 5:53
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Start with the parametrization of the torus surface: $$ x=(a+b\cos u)\cos v, $$ $$ y=(a+b\cos u)\sin v, $$ $$ z=b\sin u. $$

On that surface we get a curve that sits inside the trefoil. You get a parametrization of that curve $\vec{\gamma}=(x,y,z)$ by setting $u=3s, v=2s$ and letting $s$ range over the interval $[0,2\pi]$. This way you get a curve that wraps around the hole of the donut twice, and around the tube thrice.

Then you build a tube around that curve. You need a normalized tangent vector $$ \vec{t}(s)=\frac{\vec{\gamma}'(s)}{||\vec{\gamma}'(s)||}, $$ and a normal vector $$ \vec{n}(s)=\frac{\vec{t}'(s)}{||\vec{t}'(s)||}, $$ and a binormal vector $$ \vec{b}(s)=\vec{t}\times\vec{n}. $$

Then you can parametrize the trefoil surface $(x,y,z)=\vec{r}(s,\alpha)$ as follows $$ \vec{r}(s,\alpha)=\vec{\gamma}(s)+c\cos\alpha\vec{n}(s)+c\sin\alpha\vec{b}(s). $$ Here $c$ is the radius of the tube of the trefoil (should be smaller than $b$ = the radius of the tube of the torus). If you want elliptical cross-sections, then you use two constants in place of $c$ above (possibly phase-shifting $\alpha$). Both parameters $s$ and $\alpha$ should range over $[0,2\pi]$.

I've some Mathematica code for this somewhere, but need to dig for it. An image of the resulting trefoil (together with the torus) can be seen in the front cover of my calculus lecture notes. The trefoil is partially obscured by the torus, but that's the best version I have on-line.

Arrggh! I found my code. When producing that image I used the normal vector $$\vec{n}_T(s)=(\cos 2s\cos 3s,\sin 2s\cos 3s,\sin3s)$$ of the torus surface at the point $\vec{\gamma}(s)$ instead of the usual normal $\vec{n}(s)$. The reason may have been the simpler formula. Together with that I then also used $\vec{t}\times\vec{n}_T$ in place of the usual binormal.

I simply wanted a picture of some kind of a tube around the trefoil curve. Therefore my recipe may not be exactly what you want.

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Is the array environment not supported here? –  Jyrki Lahtonen Mar 17 '12 at 12:06
I am a little bit worried about a possible problem. It may be possible that $\vec{t}'$ vanishes at some point. IIRC if $a$ is large enough in comparison to $b$ this does not happen, but I can't exclude this possibility at the moment. We need $\vec{n}(s)$ to be a continuous function of $s$ for the parametrization to be visually pleasing. –  Jyrki Lahtonen Mar 17 '12 at 12:24
Thank you very much! –  QHLIU Mar 17 '12 at 13:52
@QHLIU, you're welcome. May I inquire as to what you're doing with this? I see you are a theoretical physicist. I recall discussing parametrization of the trefoil with a friend (he is in theoretical physics). The studied topological charges, something like Faddeev-Hopf Knots (?), and IIRC managed to get a trefoil to emerge also. I'm afraid I'm mostly ignorant about what they were doing. –  Jyrki Lahtonen Mar 17 '12 at 14:43
Found the link. –  Jyrki Lahtonen Mar 17 '12 at 20:56
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You can find an implicit parameterization of the Seifert surface using the Milnor fibration for the function $z_1^2-z_2^3$.


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