Mathematical Sciences: Knots, 3-Manifolds and Thin Position

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9704140 Thompson A fundamental question in 3-dimensional topology is the classification problem for 3-manifolds, that is, to find a complete list of 3-manifolds in which each space appears precisely once. The classification of 2-dimensional manifolds provides a model of the kind of classification sought for 3-manifolds. A crucial step in the classification problem was completed by J. H. Rubinstein in 1992, when he described an algorithm to decide if a given 3-manifold was homeomorphic to the 3-sphere, the "simplest" 3-manifold. His arguments used minimal surface theory in the context of a triangulated 3-manifold. There is a deep connection between this type of minimal surface theory and the notion of thin position for knots in the 3-sphere, an idea introduced by D. Gabai. This interplay between standard knot theory and minimal surface theory provides significant new techniques to explore outstanding problems. This project explores two particular applications of these techniques. The first is the extension of the standard 1-parameter idea of thin position for a knot to multi-parameter families. These extensions are used to approach some long-standing problems in knot theory. Connections between multi-parameter thin position and other well-known invariants, such as the energy of a knot, another notion of efficient imbedding, are also described. The second application is to a new type of irreducibility for Heegaard splittings that divides splittings into two groups, one of which is atoroidal. The techniques will be used to seek injective immersed surfaces in one group and negatively curved metrics in the other. People originally thought the earth was flat. This was a reasonable hypothesis, arrived at by considering the local information available at the time. Similarly, by considering the local information available to us, we might hypothesize that the spatial universe we live in is 3-dimensional Euclidean space. Evidence from physics makes th is unlikely. The question then is, what is the 3-dimensional shape of the universe we live in? Is it the 3-dimensional analog of the sphere? Perhaps -- but here we encounter a deep mathematical problem. Unlike 2-dimensional spaces, which are well-understood, there is no list of all 3-dimensional spaces. So we don't even understand what the possibilities are for the shape of the 3-dimensional universe. Perhaps the most important problem in low-dimensional topology today is to find such a list. This project uses the interplay between new minimization techniques from piecewise linear geometry and knot theory to explore aspects of this problem. ***