What is known about the classification of knots in a solid torus $S^1 \times D^2$? Is enumerating them a reasonable problem? Do we get a similar classification as for knots in $S^3$? Ideally there would be a simple description of the Seifert fibered knots in $S^1 \times D^2$ like for prime non-satellite knots in $S^3$ (they're exactly the torus knots).


The motivation is to better understand the classification of prime knots in $S^3$ as


1. torus knots,
2. hyperbolic knots, or
3. nontrivial satellites,


because satellites come from combining a knot in $S^3$ and a knot in $S^1 \times D^2$.


One issue is that we would want to exclude knots that are only knotted in a ball in $S^1 \times D^2$ because these aren't really new: they come from a knot in $S^3$, and they give composite knots in the satellite construction. Is there a relatively simple way to exclude these?


More formally: one class of knots to exclude are those obtained by taking a $(1,1)$-tangle in a $3$-ball, then closing up the ends so that the knot is not null-homologus in $S^1 \times D^2$. These are nontrivial knots in $S^1 \times D^2$ but they don't really use the topology of the solid torus in an interesting way, so we want to exclude them from our classification.



Classification of knots in solid torus
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.


