Mathematics Colloquia and Seminars

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It is well known that every oriented, closed 3-manifold can be obtained by Dehn surgery on a framed link in the 3-sphere. The framed link in this process is highly non-unique. Even in the relatively simpler case of 3-manifolds obtained by a surgery on a framed knot, the 3-manifold in general "remembers" little about the knot itself. In this talk we explore a new relationship between 3-manifolds resulting from surgeries on knots, and what restriction the topology of the manifold places on the knots. Specifically, we shall utilize the Heegaard Floer homology groups to establish an inequality involving the framing and the genus of the knots. Applications include an obstruction for two framed knots to yield the same 3-manifold, an obstruction that is particularly effective when working with families of framed knots. We shall introduce the rational and integral Dehn surgery genera for a rational homology 3-sphere, and use the above inequality to provide bounds, and in some cases exact values, for these genera. We also demonstrate that the difference between the integral and rational Dehn surgery genera can be arbitrarily large.