Knots, Heegaard Splittings and Width Complexes

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The proposed research grows out of a desire to better understand the workings of thin position, both in the context of knots and in the context of 3-manifolds. Simplicial complexes, cell complexes and other complexes have featured prominently in geometric group theory for many years. Their utility in the study of 3-manifolds has been established through the work of Masur and Minsky, Hempel and others. Defining and studying the ``width complex'' will help streamline current research on knot theory and Heegaard splittings. It will also reformulate some of the P.I.'s pet problems in these areas. Some of these problems pertain to the behavior of genus and rank under gluing of 3-manifolds. Others concern stabilizing of Heegaard splittings. Yet others concern the nature of untelescopings of Heegaard splittings.The proposed research grows out of a desire to better understand knots, that is, knotted circles in 3-dimensional space, and 3-manifolds, that is, 3-dimensional generalizations of surfaces. The key idea here is to use a certain type of complex, that is, a fancy bookkeeping device, to encode information about knots and 3-manifolds. Interestingly, this complex recasts standard problems of the research area in a new light. The project is extensive. Some parts are accessible to graduate students and former graduate students of the P.I. The P.I. has directed three Ph.D. dissertations to date and maintains especially close contact with Maria Robinson (Seattle University). The P.I. is currently supervising two graduate students: Shawn Lanier and Alice Stevens. The P.I. maintains several international collaborations.