Bennequin Surfaces for Links

Yi Xie, Xuanchang Liu, and Cornelia Van Cott


A link is a collection of disjoint closed curves in three-dimensional space. More intuitively, a link is a collection of several tangled pieces of string with the ends of each string glued together (see the figure above). The study of links begins with the basic question: How can we decide whether two links are the same or different? This seems easy, but the question can be extremely difficult to answer, because a simple link can be made to appear complicated by rearranging the string. To address this problem, mathematicians have developed several tools for telling links apart.

One of these tools for distinguishing links comes from another mathematical object — surfaces. A surface is a two-dimensional shape (for example, a disk, a sphere, or a plane). Every link is the boundary of many different surfaces. In our research, we focus on a special family of surfaces called Bennequin surfaces. A surface F is a Bennequin surface for link L if the boundary of F is L, F is braided (see figure below), and F has a minimal number of holes. Building on the work of several mathematicians, including Alexander Stoimenow and Mikami Hirasawa, we investigated which links bound Bennequin surfaces with only 5 disks.

Bennequin Surface