Abstract
We study the simplest class of smooth, geometrically simply-connected 4-manifolds: knot traces. Our goal is to understand which smooth structures they can support as topological manifolds, and the different ways in which surfaces can embed or fail to embed in them. We begin by studying the traces associated to shake-slice knots. We define a notion of 'complexity' for the generating spheres embedded in these traces, which is motivated by a similar phenomenon that occurs in 5-dimensional smooth h-cobordisms. We show that for each non-zero framing the minimum complexity of a smooth generating sphere in the trace of a shake slice knot can be arbitrarily high. Moreover, we give explicit upper and lower bounds on the complexity for infinite families of examples covering all non-zero framings.
We also consider exotic knot traces, which have been produced using Fintushel and Stern's knot-surgery construction. These examples atomize the construction: one can often view knot-surgery on a closed 4-manifold as knot-surgery on an embedded trace, and then study the surgered trace on its own. We explain how to find explicit embeddings of corks which undo knot surgery in any trace where the construction makes sense, modulo a parity condition which seems to have been overlooked in previous work on the topic. We obtain explicit embeddings of corks in odd degree Elliptic surfaces via this method which undo knot-surgery along a regular fiber using any companion knot. This provides a partial answer to a thirty year old question.