In 1984, T. Kobayashi gave a classification of the genus two 3-manifolds with a nontrivial torus decomposition. The intent of this study is to extend this classification to the genus two, torally bounded 3-manifolds with a separating non-trivial torus decomposition. These 3-manifolds are also known as the tunnel-1 generalized satellite knot exteriors. The main result of the study is a full decomposition of the exterior of a tunnel-1 satellite knot in an arbitrary 3-manifold. Several corollaries are drawn from this classification. First, Schubert's 1953 results regarding the existence and uniqueness of a core component for satellite knots in the 3-sphere is extended to tunnel-1 satellite knots in arbitrary 3-manifolds. Second, Morimoto and Sakuma's 1991 classification of tunnel-1 satellite knots in the 3-sphere is extended to a classification of the tunnel-1 satellite knots in lens spaces. Finally, for these knot exteriors, a result of Eudave-Muñoz in 1994 regarding the relative position of tunnels and decomposing tori is recovered.
| Identifer | oai:union.ndltd.org:pdx.edu/oai:pdxscholar.library.pdx.edu:open_access_etds-2272 |
| Date | 01 January 1995 |
| Creators | Neil, John Ralph |
| Publisher | PDXScholar |
| Source Sets | Portland State University |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | Dissertations and Theses |
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