I'm currently reading the following paper by Richard Skora, entitled Cantor sets in $S^3$ with simply connected complements found here, and on page 2, just before Theorem 1, it says "the homeomorphism $h \colon S \to S'$ is not unique even upto isotopy".
I was wondering if someone could give me a hint as to why this must be the case; I was initially wondering whether a knot or different linking in one of the genus-2-handlebodies might give rise to homeomorphisms which are not unique?