Contractible 3-manifolds and the double 3-space property

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Gabai showed that the Whitehead manifold is the union of two submanifolds each of which is homeomorphic to R³ and whose intersection is again homeomorphic to R³. Using a family of generalizations of the Whitehead Link, we show that there are uncountably many contractible 3-manifolds with this double 3-space property. Using a separate family of generalizations of the Whitehead Link and using an extension of interlacing theory, we also show that there are uncountably many contractible 3-manifolds that fail to have this property.