Unknotted solid tori and genus one Whitehead manifolds

Abstract

In this paper we study contractible open 3 3 -manifolds which are monotone unions of solid tori and which embed in a compact 3 3 -manifold. We show that the tori are unknotted in later tori. We then study pairs of unknotted solid tori, and prove a unique prime decomposition theorem. This is applied to the open 3 3 -manifolds above to get an essentially unique prime decomposition. A number of examples in the literature are analyzed, and some new examples are constructed.