Special Topology Seminar - In-Person Talk 

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Since the discovery of concordance invariants which can distinguish between topologically and smoothly slice knots, people have asked whether an exotic 4-ball, X, could be detected by finding a knot which is slice in X but not in the standard 4-ball. Many authors have shown that most of these concordance invariants are insufficiently sensitive for this strategy using algebraic or geometric arguments. We will explore the viability of this strategy using purely topological methods. We will give a general method for constructing an embedding of the 2-skeleton of X into the standard 4-ball. When this is possible, it implies that the smooth slice genus of knots in X must be equal to their genera in the 4-ball. It is not yet clear how general this method is, perhaps it may work for all homotopy 4-balls, but we conjecture that it is sufficient for Gluck twists. We welcome any comments or suggestions!