Geometry and Topology Seminar

Suppose that L is a 2-component link with an integral Dehn surgery yielding the connected sum of two copies of S^1 x S^2. How complicated can L be? Considerations of this sort fall under the purvey of the Generalized Property R Conjecture, which we will broadly overview in this talk, drawing connections with a number of open problems in low-dimensional topology. In particular, we will give many new potential counterexamples to the stronger versions of the GPRC (with relevance to the Andrews-Curtis Conjecture and the Slice-Ribbon Conjecture), while also giving an infinite family of knots that can never occur in a 2-component counterexample to the weakest version of the GPRC (with relevance to the Poincaré Conjecture). This joint work with Alex Zupan.