Mathematical Sciences: Low Dimensional Manifolds and Knot Theory

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9626550 Gordon Cameron Gordon will continue to investigate Dehn surgery on knots. The main goal is further to circumscribe the various exceptional (non-hyperbolic) surgeries on hyperbolic knots. The known bounds on the number and nature of such surgeries are now quite sharp in many cases, and it seems that it may eventually be possible (at least modulo the Geometrization Conjecture of Thurston) to give an essentially complete description of all exceptional surgeries. The methods used will be based largely on the combinatorial- topological techniques developed by Gordon and John Luecke in their previous work on Dehn surgery, such as the proof of the Knot Complement Conjecture. This project deals with knot theory and 3-dimensional topology, the general goal of the latter being to understand the structure of 3-dimensional manifolds. These are objects that are locally like ordinary 3-dimensional Euclidean space but whose global structure may be quite complicated. Since we live in a 3-manifold, one might say that 3-dimensional topology aims to describe what the mathematical possibilities are for our spatial universe. This aim is still not realized, although there is tantalizing evidence that such a description might ultimately be possible. One important aspect of 3-dimensional topology is the theory of knots---a knot being a closed loop embedded somehow in ordinary 3-dimensional space. On the one hand, results about 3-manifolds often give information about knots as special cases, while on the other hand, a wide variety of mathematical methods can be applied to the study of knots, leading in turn to new information about 3-manifolds. (Recently, deep connections between 3-dimensional topology and quantum field theory were discovered in this way.) The main vehicle by which knot theory relates to the general theory of 3-manifolds is Dehn surgery, and this is the focus of the current project. In Dehn surgery, a solid tube around a knot is removed and "s ewn back" differently, giving a new 3-manifold. If one allows links (i.e., several loops linked together) as well as knots, then every 3-manifold can be obtained in this way, so a sufficiently good understanding of this construction would have important implications for the general theory of 3-manifolds. ***