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The answer to the question, as asked, is "no". For, suppose that $B$ is a three-ball. So, $B$ is a genus zero handlebody. Let $\alpha$ be a knotted arc properly embedded in $B$. So the fundamental group of $B - \alpha$ is not free. Let $A$ be the frontier of a regular neighbourhood of $\alpha$. So $\alpha$ is an annulus, properly embedded in $B$. Cutting $B$ along $A$ and taking closures yields a three-ball ("inside $A$") and a knot complement ("outside $A$").