Dehn Surgery on Links

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9803122Luecke This project aims to understand the topology of the exterior of ahyperbolic knot that admits a non-hyperbolic Dehn surgery. Morespecifically, what are the hyperbolic knots in the 3-sphere that admitDehn surgeries that are small Seifert fibered spaces or contain essential2-spheres or 2-tori? The main approach to this problem is to understandwhen a surface of small genus, either an essential surface or a Heegaardsurface, can appear in a Dehn surgery on a hyperbolic knot. The residueof a surface in the Dehn surgery is seen as a punctured surface in theknot exterior. It is put forth that by looking at the intersections ofsuch a punctured surface with an appropriate planar surface in the knotexterior, one will obtain a good picture of the knot. Such an approachhas been successful in the past at analyzing similar situations. The structures of 3-dimensional spaces and the classification of knotsare beautiful and deep subjects, but they are also important subjects inextending our knowledge in other branches of the sciences. For example,some physicists now are trying to tie astronomical observations tostructures of 3-dimensional spaces in order to determine the shape of the3-dimensional universe. Microbiologists are beginning to use the conceptsof knot theory to describe and understand the knotting of DNA molecules.Chemists are doing the same to understand the knotting and linking of othermolecules. The Dehn surgery construction is an operation on a 3-dimensionalspace that produces a new 3-dimensional space. This is done by changingthe space around a knotted circle in the space. This project studies howthe space changes according to the type of knot along which the change wasmade. The relationship between the resulting space and the knot is acomplex one, and its understanding is a natural goal for 3-dimensionaltopology and knot theory. A good knowledge of this relationship is also auseful tool for exploring knots and 3-dimensional spaces. Since itsintroduction at the turn of the century, the Dehn surgery construction hasbeen a standard way of constructing examples of phenomena in 3- and4-dimensional spaces. It also turns out that many basic questions aboutknots can be formulated as questions about the Dehn surgery construction. ***