Unknotted Solid Tori and Genus One Whitehead Manifolds

Abstract

In this paper we study contractible open $3$-manifolds which are monotone unions of solid tori and which embed in a compact $3$-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 $3$-manifolds above to get an essentially unique prime decomposition. A number of examples in the literature are analyzed, and some new examples are constructed.