Braids and Contact Geometry

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The principal Investigator (PI) plans to apply the theory of geometric braids to study a variety of problems in contact geometry. Especially, she introduces a new notion, open book foliation, that has its origin in braid theory. She intends to apply the open book foliation to investigate contact structures for three dimensional manifolds and find new tightness and overtwistedness criteria. In addition, the PI continues to study the relationship between the braid index, a classical knot invariant, and various geometric quantities such as writhes, HOMFLY polynomials, the self-linking number in contact geometry, and Khovanov-Rozansky homology. The PI will disseminate the results of her research by advising her current and potential graduate students at the University of Iowa, and by interacting with both undergraduate and graduate students through her seminars and lectures. The projects afford undergraduate and graduate students the opportunity to study contact geometry from both a theoretical and practical point of view. Developing computer graphic techniques with a view towards visualizing the open book foliations will provide students with an excellent introduction to a modern mathematicsThe PI will work on several projects in low-dimensional topology. Low dimensional topology is broadly applicable in a variety of scientific disciplines including biology and medicine where, for example, protein folding mechanisms and DNA structure play a seminal role. Results of the proposed projects will likely stimulate communication and collaboration between the PI and researchers within mathematics as well as other scientific disciplines.