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Marty Scharlemann
A 3-dimensional reimbedding strategy for the 4-dimensional Schoenflies Conjecture

A proof of the generalized Property R Conjecture in dimension 3 could be useful in proving the Schoenflies Conjecture in dimension 4. Part of the proposed strategy requires a Fox type reimbedding of a 3-dimensional cross-section M^3 \subset S^3 x {0} \subset S^4 in a usefully natural way. For example, if M lies inside a knotted solid torus, a natural reimbedding of M would be to unknot the solid torus in which lies. Sadly, when M lies inside a higher genus knotted handlebody W, there isn't such a natural choice of reimbedding. But for the special case in which W is gotten by tubing together two distant solid tori there is a natural choice: both unknot the tori and straighten the tube. We show how this case alone suffices to prove the genus three, 4-dimensional Schoenflies Conjecture.