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Thin position, bridge structure, and monotonic simplification of knots

Since its inception, the notion of thin position has played an important role in low-dimensional topology. Thin position for knots in the 3-sphere was first introduced by David Gabai in order to prove the Property R Conjecture. In addition, this theory factored into Cameron Gordon and John Luecke’s proof of the knot complement problem and revolutionized the study of Heegaard splittings upon its adaptation by Martin Scharlemann and Abigail Thompson. Let h : S → R be a Morse function with two critical points. Loosely, thin position of a knot K in S is a particular embedding of K which minimizes the total number of intersections with a maximal collection of regular level sets, where this number of intersections is called the width of the knot. Although not immediately obvious, it has been demonstrated that there is a close relationship between a thin position of a knot K and essential meridional planar surfaces in its exterior E(K). In this thesis, we study the nature of thin position under knot companionship; namely, for several families of knots we establish a lower bound for the width of a satellite knot based on the width of its companion and the wrapping or winding number of its pattern. For one such class of knots, cable knots, in addition to finding thin position for these knots, we establish a criterion under which non-minimal bridge positions of cable knots are stabilized. Finally, we exhibit an embedding of the unknot whose width must be increased before it can be simplified to thin position.