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Lisa Piccirillo, RLM 12.166: The Conway knot is not slice: applications of a flexible construction of knots with diffeomorphic traces
Monday, September 17, 2018, 02:00pm - 03:00pm
The surgery-theoretic classification of manifolds relies on findingembedded sphere representatives for each middle dimensional homologyclass. The fact that in 4-manifolds not all second homology classeshave smooth sphere representatives is then key to the complexityof smooth 4-manifold topology. The simplest 4-manifolds wherethis occurs, knot traces, are of fundamental importance. I'llgive a flexible technique for constructing pairs of distinctknots with diffeomorphic traces. Using this construction, I willshow that there are knot traces where the minimal genus smoothsurface generating second homology is not the obvious one, resolvingquestion 1.41 on the 1978 Kirby problem list. I will also usethis construction to show that Conway knot does not bound a smoothdisk in the four ball, which completes the classification ofslice knots under 13 crossings and gives the first example ofa non-slice knot which is both topologically slice and a positivemutant of a slice knot.
Location: RLM 12.166

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