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I am not very familiar with knot theory nor with minimal surfaces, so I already apologize if my question appears too naive or simple :). I am trying to do the following:

Starting from a real trigonometric parameterization for the trefoil knot:

$$ \begin{array}{lcl} x(\theta) &=& \cos(\theta) - 2 \cos(2\theta) \\ y(\theta) &=& \sin(\theta) + 2 \sin(2\theta) \\ z(\theta) &=& 2\sin(3\theta) \end{array} $$

I am hoping to deduce a Enneper-Weierstrass Parameterization (or at least isothermal) so I am naively complementing imaginary parts to the parametric expressions above in order to deal with analytic meromorphic functions. The real parts of each function corresponds to the trefoil (x,y,z) real coordinates. i.e. Naturally I am tempted to write:

\begin{array}{lcl} \Gamma_1(z) &=& z - 2 z^2 \\ \Gamma_2(z) &=& -i(z + 2 z^2) \\ \Gamma_3(z) &=& -2i z^3 \end{array}

with

\begin{array}{lcl} \Re\left\{\Gamma_1(z)\right\} &=& x(\theta) \\ \Re\left\{\Gamma_2(z)\right\} &=& y(\theta) \\ \Re\left\{\Gamma_3(z)\right\} &=& z(\theta) \end{array}

So with,

\begin{array}{lcl} \frac{\partial \Gamma_1(z)}{\partial z} &=& \Phi_1(z) \\ \frac{\partial \Gamma_2(z)}{\partial z} &=& \Phi_2(z) \\ \frac{\partial \Gamma_3(z)}{\partial z} &=& \Phi_3(z) \end{array}

One would have an isothermal parameterization of the trefoil minimal surface if

$$\Phi_1(z)^2+\Phi_2(z)^2+\Phi_3(z)^2 = 0$$

Which, clearly does not hold...

So how could I further progress? Apparently, an analytic parameterization for a surface does define a surface, but not necessarily the minimal surface. However the minimal surface certainly exists and is unique. Its isothermal parameterization is very likely to coincide on the boundary (the trefoil) with the one we already have here (at least for the real part...).

Any idea in order to further proceed and come up with a parametarization for the trefoil minimal Seifert Surface?

Thanks a lot in advance,

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  • $\begingroup$ Have you looked at the Milnor fibration? I've never checked to see if it defines minimal surfaces but presumably that is known. A quick Google should lead you to it. $\endgroup$ – Ryan Budney May 17 '16 at 2:10
  • $\begingroup$ Hi Ryan, Thanks for your suggestion. Indeed Milnor fibration seems to be a promising strategy, according to google searches I made. My ultimate goal however is to derive an analytic equation for the trefoil knot Seifert surface. I have to say google search so far did not reveal much details about the principles behind Milnor Fibration, such that I still do not know how to proceed. Any good reference on minor fibration and trefoil knot Seifert surface parameterization? $\endgroup$ – handelskai May 17 '16 at 8:34

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