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
- torus knots,
- hyperbolic knots, or
- 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.