The central theme for this paper is provided by the following three statements: (1) Every compact connected 1-manifold is S1. (2) Every compact connected simply connected 2-manifold is S2. (3) Every compact connected simply connected 3-manifold is S3. We provide proofs of statements (1) and (2). The veracity of the third statement, the Poincaré Conjecture, has not been determined. It is known that should a counter-example exist it can be found by removing from S3 a finite collect ion of solid tori and sewing them back differently. We show that it is not possible to find a counterexample by removing from S3 a single solid torus of twist knot type or torus knot type and sewing it back differently. We treat as special cases a solid torus of trivial knot type and trefoil knot type.