Let $M^3$ be some compact submanifold of $S^3$ with connected boundary. I am interested in the failure of the analog of Waldhausen's theorem for $M^3$ - namely, I would like examples of such $M$ together with two Heegaard splitting surfaces in $M$ that are not isotopic.
I thought, I remembered reading about such examples (maybe even with the trefoil complement as $M$) maybe with respect to some "exotic unknotting tunnels", but it could all be a dream.
