This project aims to understand the topology of the exterior of a hyperbolic knot that admits a non-hyperbolic Dehn surgery. More specifically, what are the hyperbolic knots in the 3-sphere that admit Dehn surgeries that are small Seifert fibered spaces or contain essential 2-spheres or 2-tori? The main approach to this problem is to understand when a surface of small genus, either an essential surface or a Heegaard surface, can appear in a Dehn surgery on a hyperbolic knot. The residue of a surface in the Dehn surgery is seen as a punctured surface in the knot exterior. It is put forth that by looking at the intersections of such a punctured surface with an appropriate planar surface in the knot exterior, one will obtain a good picture of the knot. Such an approach has been successful in the past at analyzing similar situations. The structures of 3-dimensional spaces and the classification of knots are beautiful and deep subjects, but they are also important subjects in extending our knowledge in other branches of the sciences. For example, some physicists now are trying to tie astronomical observations to structures of 3-dimensional spaces in order to determine the shape of the 3-dimensional universe. Microbiologists are beginning to use the concepts of knot theory to describe and understand the knotting of DNA molecules. Chemists are doing the same to understand the knotting and linking of other molecules. The Dehn surgery construction is an operation on a 3-dimensional space that produces a new 3-dimensional space. This is done by changing the space around a knotted circle in the space. This project studies how the space changes according to the type of knot along which the change was made. The relationship between the resulting space and the knot is a complex one, and its understanding is a natural goal for 3-dimensional topology and knot theory. A good knowledge of this relationship is also a useful tool for exploring knots and 3-dimensional spaces. Since its introduction at the turn of the century, the Dehn surgery construction has been a standard way of constructing examples of phenomena in 3- and 4-dimensional spaces. It also turns out that many basic questions about knots can be formulated as questions about the Dehn surgery construction. ***
