Mathematics Colloquia and Seminars

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Sutured manifold theory is one of the primary advances in 3-dimensional topology to come out of the 1980's. First developed by David Gabai, who used it to solve a number of long-standing problems in knot theory, it has continued to find a wide variety of applications. In this talk, I will discuss an application of this theory to the study of links via the orientable surfaces that they bound in 3-space, known as Seifert surfaces. The focus will be on an important class of links called fibered links, whose complements in the 3-sphere are fiber bundles over the circle. A fiber of such a bundle is naturally called a fiber surface for the link. After discussing these links, I will illustrate sutured manifold theory by using its basic elements to show that an important method of gluing two Seifert surfaces S and S' together (called the Murasugi sum) produces a fiber surface if and only if both S and S' are fiber surfaces. Everything will be defined from scratch, so no deep knowledge of knot theory will be required. This talk is related to the topic to be discussed by Dorothy Buck afterward, and should serve as an introduction to most of the relevant concepts.