Mathematics Colloquia and Seminars

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We will introduce some important ideas from both knot theory and the theory of Heegaard splittings, then describe a useful connection between them. In particular, we will show that the bridge number of a non-trivial, tunnel-number-one knot is bounded below by a certain distance in a graph called the curve complex. From this, and a technical lemma about the curve complex, it follows that there are (hyperbolic) tunnel-number-one knots with arbitrarily high bridge number. The talk should be accessible to beginning grad. students. All the terms used in the abstract will be defined.