An infinite family of links with critical bridge spheres

A 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 multi- dimensional 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.