Knot traces and the slice genus

Knot traces are elementary 4-manifolds built by attaching a single 2-handle to the 4-ball; these are the canonical examples 4-manifolds with non-trivial middle dimensional homology. In this thesis, we give a flexible technique for constructing pairs of distinct knots with diffeomorphic traces. Using this construction, we show that there are knot traces where the minimal genus smooth surface generating homology is not the canonical surface, resolving a question on the 1978 Kirby problem list. We also use knot traces to give a new technique for showing a knot does not bound a smooth disk in the 4-ball, and we show that the Conway knot does not bound a smooth disk in the 4-ball. This resolves a question from the 1960’s, 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.Mathematic