# The Conway knot is not slice

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Lisa Piccirillo, UT Austin
Fine Hall 314

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. I’ll 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 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.