Combinatorial and Geometric Problems in Knot Theory

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Proposal: DMS-9971244PI: Morwen ThistlethwaiteAbstract: Thistlethwaite's research is in classical knot theory, the study of embeddings of smooth simple closed curves in the 3-sphere. The computer is an essential tool in his research, in that the problems he addresses are often suggested by careful observation of data. Thistlethwaite will continue to investigate the rate of growth of the number of knots, following his collaboration with C. Sundberg, where the exact growth exponent was determined for prime, alternating links. He will study optimal configurations of symmetric knots, and will investigate other geometric problems associated with knot complements, in particular hyperbolic knot complements containing essential4-punctured spheres. He will investigate the possibility that hyperbolic geometry might be used to provide a new, completely geometric proof of the Tait flyping conjecture. The knot tables will be extended well beyond the 1,701,936 knots currently listed, and these new tables will be used to search for interesting examples. Development will continue on the freely available software package "Knotscape", which provides a graphical interface to the knot tables and access to many invariants.Thistlethwaite's specialty is classical knot theory, a branch of 3-dimensional topology with origins in the nineteenth century. A knot is a closed curve in 3-dimensional space; one can imagine a knotted rope with its ends joined together. Two knots are considered to be equivalent if one can be deformed continuously to the other, and a knot is deemed to be trivial, or unknotted, if it is equivalent to a flat circle. Knot theory is a rich subject, as it interfaces with geometry, topology, algebra and combinatorics. In recent years it has aroused the interest of chemists and molecular biologists, particularlyin relation to the knotting of DNA. Knot theory abounds with problems which are simple to state, but hard to solve; for example, it is often difficult in practice to prove that two given knots are inequivalent, as it is hard to rule out the existence of some ingenious method of deforming one to the other. It can even be hard to decide whether a given knot is trivial. If a knot is laid down on a flat surface with as few crossovers as possible, the resulting number of crossovers is called the crossing-number of the knot. To date, Thistlethwaite has classified by computer the 1,701,936 knots of up to 16 crossings; this classification was confirmed by an independent tabulation carried out by J. Hoste and J. Weeks. As part of this project, he will extend the tables to 17 or 18 crossings, thereby expecting to find many new examples with exciting properties. He will continue to investigate the fundamental problem as to how fast the number of knots grows in relation to crossing-number. Thistlethwaite will use hyperbolic geometry, a form of non-Euclidean geometry, to investigate hidden symmetries of knots, and to determine the structure of alternating knots (a knot is alternating if it can be arranged so that the rope goes alternately over and under at successive crossings.) He will continue to develop the software package "Knotscape", which provides a graphical interface to the knot tables.