An infinite family of links with critical bridge spheres

Abstract

ACKNOWLEDGEMENTSThis thesis represents the work, support, encouragement, and love of countless family, friends, mentors, teachers, and pastors, each responsible for a piece of my development as a learner, a teacher, and a human being. Though it is tempting to try, it would be impossible to recognize every one of them here, so I will have to limit these acknowledgments to those individuals who have specifically shaped me as a mathematician. Soli Deo gloriaii ABSTRACT A closed, orientable, splitting surface in an oriented 3-manifold is a topologically minimal surface of index n if its associated disk complex is (n−2)-connected but not (n − 1)-connected. A critical surface is a topologically minimal surface of index 2. In this thesis, we use an equivalent combinatorial definition of critical surfaces to construct the first known critical bridge spheres for nontrivial links.iii PUBLIC ABSTRACTA knot can be thought of as the result of taking a shoe string, twisting it up in some way, and gluing the ends together. A link is a set of any number of knots which may be twisted and wound up around each other. When a sphere intersects a link in such a way that the link is cut into pieces such that each piece has exactly one maximum or minimum point, the sphere is said to be a bridge sphere for the link.Making extensive use of a tool called a compressing disk, we can study how a link and its bridge sphere intersect each other by associating to them a multidimensional space Γ called a simplicial complex. A natural question is, does Γ have any 2-dimensional holes in it? In other words, is every circle in Γ the boundary of a disk? If Γ is connected, but it contains a 2-dimensional hole, then the bridge sphere is called critical.We use an equivalent and more tangible definition of a critical bridge sphere defined by how compressing disks on one side of the sphere intersect compressing disks on the other side. In this thesis, we study an infinite family of links which can be laid out in a specific plat pattern. We examine their bridge spheres and how their compressing disks intersect each other to conclude that each of these bridge spheres is critical.