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…)
Seifert Surfaces of Torus Knots
gt.geometric-topologyknot-theoryseifert-surfaces
Related Solutions
A "long topological knot" in $\mathbb R^n$ is a topological embedding $f : \mathbb R \to \mathbb R^n$ such that $f(x) = (x,0)$ for all $x \in \mathbb R \setminus (-1,1)$.
Let $K_n$ be the space of all long topological knots in $\mathbb R^n$ with the compact-open topology. Then $K_n$ is contractible. The contraction is given by
$F : [0,1] \times K_n \to K_n$ defined by
$F(t,f)(x) = (1-t)f(\frac{x}{1-t})$ provided $t \in [0,1)$ and $F(1,f)(x) = (x,0)$.
This map, $F$, is sometimes called "The Alexander Trick". See the Wikipedia Alexander Trick page for context.
In response to your edit, perhaps an interesting topology on $K_n$ could be given this way. Given $f \in K_n$ and $\epsilon > 0$ we'll say an $\epsilon$-ball about $f$ consists of all knots $\phi \circ f$ where $\phi : \mathbb R^n \to \mathbb R^n$ is a homeomorphism which agrees with the identity map outside of $D^n$, and such that $|\phi(x)-x|<\epsilon$ for all $x \in \mathbb R^n$. The topology on $K_n$ could be the topology generated by all $\epsilon$-balls about all $f \in K_n$. Presumably this kind of topology has a name?
Up to Dehn twists, the class of knots in the solid torus is identical to the class of two-component links in the three-sphere, where the first component is an unknot.
For example, Seifert fibered knots in the solid torus give Seifert fibered links in the three-sphere. The base space is the orbifold $S^2(p,\infty,\infty)$.
There is a similar theory of “satellite” links. The knots you wish to exclude (knotted in a three-ball inside the solid torus) are a special case of these. When thought of as links, they are exactly the split links in the three-sphere (where again the first component is the unknot).
Finally, the knot complement in the solid torus is hyperbolic exactly when the link complement in the three-sphere is.
Best Answer
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.