Seifert surfaces of torus knots

Does anyone know a nice description of a Seifert surface of a torus knot? I can construct such surfaces in band projection, but what I get is ugly and unwieldy. Is there some elegant description for Seifert surfaces for such knots which I'm missing? (I'm not sure precisely what I mean by elegant...)

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There's the usual description of the Seifert surface for a general cable obtained by taking copies of a Seifert surface for the knot and a fiber for the cable in the solid torus. See Ken Baker's discussion.

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This picture--for a (7,2) torus knot--shows a geometric pattern you can extend to any (n,2) torus knot.

The image is part of Figure 18 in the visually rich paper by Jarke van Wijk and Arjeh Cohen: "Visualization of the Genus of Knots." Proceedings IEEE Visualization 2005.

You can even download the SeifertView software they used to generate the picture.

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Beautiful! Gets my +1. –  Joel Fine Nov 25 '09 at 16:20
Torus knot complements fiber over $S^1$. So the minimal Seifert surface for a $(p,q)$-torus knot is a once punctured surface of genus $\frac{(p-1)(q-1)}{2}$. You get it as the Milnor fibre of the map from $\mathbb C^2 \to \mathbb C$ given by $f(z_1,z_2)=z_1^p-z_2^q$.
That's pretty elegant to me. The monodromy is an automorphism of the surface of order $pq$, it is a free action except on two orbits -- one orbit has $p$ elements, the other orbit has $q$ elements.