2:00 pm Monday, October 12, 2015
Junior Topology: Families of Knots Admitting the Same Surgery by Lisa Piccirillo (UT Austin) in RLM 12.166
A classical problem in Dehn surgery asks whether there exist homology three spheres, or any three manifolds, which can be obtained by surgery on infinitely many distinct knots. In '98 Cooper and Lackenby showed that for highly non-integral surgery on hyperbolic knots, such manifolds do not exist. But in 2006 Osinach gave a method of constructing infinite families of distinct knots which have integer surgeries resulting in the same three manifold (in 2014, joint with Abe, Jong, and Luecke, homology three sphere), and these examples include families of hyperbolic knots. This year, Yasui gave a different method of constructing (finite) families of knots admitting the same surgery. We'll talk about these constructions, their use in the resolution of the Akbalut-Kirby conjecture, and their relevance to slice-ribbon and smooth four dimensional Poincare conjectures. If time, we will also talk about some results regarding extending these surgery homeomorphisms across the trace of the surgery. This talk should only assume differential topology prelim material. This is a Junior Topology talk. Submitted by
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