The main focus of the work is in the area of knot theory, specifically the class of tunnel number 1 knots, or equivalently the knots whose exteriors admit genus-2 Heegaard splittings. These include many of the common types of knots, such as 2-bridge knots, torus knots, and genus-1 1-bridge knots. The work already underway gives a new theoretical description of this class, by relating it to combinatorial constructions originating in group theory and the theory of mapping class groups of handlebodies. This description yields a unique procedure to construct any knot tunnel, and even a numerical parameterization of all the tunnels of all tunnel number 1 knots. It provides the foundation for a new level of investigation in this area, with many directions being pursued in ongoing research. Additional work with several other investigators will examine questions about minimal triangulations of 3-manifolds, and at least in its initial stages will develop software to examine large collections of examples.
Because of its connections with numerous other mathematical areas, and its relevance to the 3-dimensional space in which we live, 3-dimensional topology has been a vigorous area of research for many decades. The study of knots is one of its central themes, and reflects this rich diversity of viewpoints. The work underway develops new connections between tunnel number 1 knots and certain disk complexes and curve complexes, which are objects of much recent interest in low-dimensional topology and Teichmuller theory. As basic research in a pure theoretical discipline, the work does not envision immediate applications to science or technology. Nonetheless, there are numerous ways in which the PI's ongoing research program has a broader impact in education and student research. The work to date and its planned continuations involve heavy participation by the PI's doctoral students. The PI served for seven years as director of his department's graduate program, in particular stressing the recruitment of women and minorities into graduate-level mathematics. The PI also directs undergraduate research students, and has long been active in regional activities of the Mathematical Association of America, including service as Section Governor.
