What is parameterization of the trefoil knot surface in R³?

What is a parameterization, say (x(u,v),y(u,v),z(u,v)), of the trefoil knot surface in R³ whose cross-section can be circular or, in general, elliptic?

Thanks!

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You mean the Seifeit surface? – Kerry Mar 17 '12 at 5:53
Or did you mean to say curve, and have a single-variable function? – Ryan Budney Aug 15 '14 at 17:50

2 Answers

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 $[0,2\pi]$.

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. Anyway, here is the pic of the resulting tube around the trefoil on the surface of a torus:

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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
@Jyrki Lahtonen Trefoil knot surface in R³ seems quite inevitable in physics world, e.g. in the string theory, electromagnetism, fluid membrane, condensed matter physics, nanostructure, ..., cf. Faddeev model (hopfion.com/faddeev.html), Linked and knotted beams of light (nature.com/nphys/journal/v4/n9/abs/nphys1056.html), etc. What I am interested if quantum constrained motion on the surface possibly results in some novel consequence, cf. Geometric momentum: The proper momentum for a free particle on a two-dimensional sphere (pra.aps.org/abstract/PRA/v84/i4/e042101). – QHLIU Mar 18 '12 at 0:26

You can find an implicit parameterization of the Seifert surface using the Milnor fibration for the function $z_1^2-z_2^3$.

http://en.wikipedia.org/wiki/Milnor_map

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