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Caltech/USC/UCLA Joint Topology Seminar

Monday, October 15, 2018
5:00pm to 6:00pm
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The Conway knot is not slice
Lisa Piccirillo, Department of Mathematics, The University of Texas at Austin,

UCLA, MS 6627

Surgery-theoretic classifications fail for 4-manifolds because many 4-manifolds have second homology classes not representable by smoothly embedded spheres. Knot traces are the prototypical example of 4-manifolds with such classes. Ill give a flexible technique for constructing pairs of distinct knots with diffeomorphic traces. Using this construction, I will show that there are knot traces where the minimal genus smooth surface generating second homology is not the obvious one, resolving question 1.41 on the 1978 Kirby problem list. I will also use this construction to show that Conway knot does not bound a smooth disk in the four ball, which completes the classification of slice knots under 13 crossings and gives the first example of a non-slice knot which is both topologically slice and a positive mutant of a slice knot.

For more information, please contact Mathematics Department by phone at 626-395-4335 or by email at [email protected].