Higher Dimensional Convex Brunnian Links and Other Explorations in Knots
Higher Dimensional Convex Brunnian Links and Other Explorations in Knots. Thesis under the direction of Hugh N. Howards, Ph.D., Associate Professor of Mathematics. This thesis will examine two areas of knot theory. The first involves stick numbers of knot, or the minimum number of straight line segments required to make a knot. The triple products of three consecutive sticks can be right or left-handed, which is an important property to the structure of a stick representation of a knot. We will examine a few ways to construct a stick representation of a knot so that all the vectors in the stick representation are right-handed, or all the vectors in the stick representation are left-handed. We will finish this area with a newly discovered lower bound on the number of sticks it required to make a knot out of entirely right or left-handed vectors. The second area involves convex Brunnian links and their analogs in higher dimensions. Using a proof constructed in collaboration with Dr. Hugh Howards, Mr. Robert Davis, and Dr. Jason Parsley, we demonstrate that certain higher dimensional knots are indeed convex Brunnian links. This will lead to the construction of an infinite family of these convex Brunnian links. We will also examine a special case of convex Brunnian links and provide a proof that these are convex Brunnian links.
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