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In his 3-manifolds notes (page 11, item $(4)$), Hatcher shows that a 2-sided compressible torus $T$ in an irreducible 3-manifold $M$ either bounds a solid torus $S^1 \times D^2$ or is contained in a ball $B \subset M$.

Is it then correct to say that such a torus in a manifold meeting those conditions must always bound a solid torus? I believe (maybe I'm mistaken) that any torus inside a solid ball should bound a solid torus. Thanks!

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You are indeed mistaken.

For a counterexample, start with any nontrivial knot $K \subset \mathbb R^3 \subset S^3$.

Let $N \subset \mathbb R^3 \subset S^3$ be a solid torus neighborhood of $K$, with boundary torus $T$.

Let $B \subset \text{interior}(N)$ be a solid ball embedded in the interior of $N$ and thus disjoint from $T$.

Then $$B^{c} = S^3 - \text{interior}(B) $$ is a solid ball into which $T$ is embedded, but $T$ does not bound a solid torus in $B^c$.

Examples like this are sometimes called "knotted holes" or "knotted hole balls". Here's a link depicting the knotted hole ball associated to a trefoil knot.

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  • $\begingroup$ Ahh, I see, thank you very much! $\endgroup$
    – Mauro
    May 17, 2022 at 20:07

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