5 TeX it up

Torus knot complements fiber over S^1. $S^1$. So the minimal Seifert surface for a (p,q)-torus $(p,q)$-torus knot is a once punctured surface of genus (p-1)(q-1)/2. $\frac{(p-1)(q-1)}{2}$. You get it as the Milnor fibre of the map from $\mathbb C^2 --> C \to \mathbb C$ given by f(z_1,z_2)=z_1^p-z_2^q. $f(z_1,z_2)=z_1^p-z_2^q$.

That's pretty elegant to me. The monodromy is an automorphism of the surface of order pq, $pq$, it is a free action except on two orbits -- one orbit has p $p$ elements, the other orbit has q $q$ elements.

These details are mostly in Milnor's "Singular points of complex hypersurfaces", also Eisenbud and Neumann's "Three-dimensional link theory and invariants of plane curve singularities". I also have a sketch of it in my JSJ-decompositions paper, on the arXiv. I got the idea for this computation by fleshing out an example of Paul Norbury's (from Walter Neumann's canonical decompositions paper on his webpage).

I'd like to add, Eisenbud and Neumann describe the Seifert surfaces of all knots whose complements are graph manifolds in this way. Well, the ones that fibre. They also characterise the knots with graph manifold complements that fiber over S^1.

edit: alternatively you could construct the Seifert surface and monodromy from the Seifert-fiber data, as I sketch in this thread: http://mathoverflow.net/questions/7746/periodic-mapping-classes-of-the-genus-two-orientable-surface/7747#7747

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 (p-1)(q-1)/2. You get it as the Milnor fibre of the map from C^2 --> 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.

These details are mostly in Milnor's "Singular points of complex hypersurfaces", also Eisenbud and Neumann's "Three-dimensional link theory and invariants of plane curve singularities". I also have a sketch of it in my JSJ-decompositions paper, on the arXiv. I got the idea for this computation by fleshing out an example of Paul Norbury's (from Walter Neumann's canonical decompositions paper on his webpage).

I'd like to add, Eisenbud and Neumann describe the Seifert surfaces of all knots whose complements are graph manifolds in this way. Well, the ones that fibre. They also characterise the knots with graph manifold complements that fiber over S^1.

edit: alternatively you could construct the Seifert surface and monodromy from the Seifert-fiber data, as I sketch in this thread: http://mathoverflow.net/questions/7746/periodic-mapping-classes-of-the-genus-two-orientable-surface/7747#7747

3 more details

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 (p-1)(q-1)/2. You get it as the Milnor fibre of the map from C^2 --> 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.

These details are mostly in Milnor's "Singular points of complex hypersurfaces", also Eisenbud and Neumann's "Three-dimensional link theory and invariants of plane curve singularities". I also have a sketch of it in my JSJ-decompositions paper, on the arXiv. I got the idea for this computation by fleshing out an example of Paul Norbury's (from Walter Neumann's canonical decompositions paper on his webpage).

I'd like to add, Eisenbud and Neumann describe the Seifert surfaces of all knots whose complements are graph manifolds in this way. Well, the ones that fibre. They also characterise the knots with graph manifold complements that fiber over S^1.

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