Mathematics Colloquia and Seminars

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Fibered knots form a ubiquitous class of knotted curves which have been actively studied since the 1960's. I will provide an introduction to their study, beginning with fundamental examples and an overview of ways to determine if a knot is fibered. Two important ways of constructing new fibered knots from known ones (Murasugi sums and Stallings twists) will be described, and I will discuss a subtle, recently-discovered issue that arises in using the second operation in practice. Throughout, a major theme will be how the study of the 3-dimensional complement of a fibered knot in S^3 can often be played down to two dimensions via the monodromy map. If time allows, I will describe the connection of fibered knots to two major open problems in low-dimensional topology.