# Overview

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**Lecture 1: Tuesday September 10, 4pm, PK-5675**

## Diffeomorphic knot traces

*Abstract*: The trace embedding lemma states that a knot K is slice if and only if X0(K) embeds smoothly in the 4-ball. This powerful observation, alongside minor contributions from Freedman and Donaldson, yields an elegant proof that R4 supports multiple smooth struc- tures. In hopes of other applications, it becomes natural to ask whether knot traces (or more generally whether any manifolds naturally associated to a knot K) determine the concordance class of K. Iâ€™ll survey the literature on this subject. Then, Iâ€™ll outline how to use the failure of knot traces to determine concordance to give a sliceness obstruction, and show that the Conway knot is not slice. I will also discuss strategies for making this proof outline more broadly applicable, including potentially to bounding smooth 4-genus and to problems in topological concordance.

**Lecture 2: Wednesday September 11, 4pm, PK-5675**

## Exotic knot traces

*Abstract*: A smooth 4-manifold X is exotic if there exists another smooth 4 manifold W which is homeomorphic but not diffeomorphic to X. One of the primary goals of smooth 4-manifold topology is to build exotic compact closed manifolds with simple algebraic topology. From a handlebody-theoretic perspective, the simplest compact, contractible 4-manifolds with boundary are the Mazur manifolds. I will use exotic knot traces to produce exotic Mazur manifolds. The diffeomorphism obstruction will come from proving that the knot Floer homology concordance invariant Î½ is a knot trace invariant. I will also give an application to a generalization of property R and discuss how exotic knot traces might be used to produce exotic homotopy 4-balls. Much of the work discussed in this lecture is joint with Kyle Hayden and Tom Mark.

**Lecture 3: Thursday September 12, 4pm, PK-5675**

## Not knot traces

*Abstract*: A smooth 4-manifold is a not knot trace if it is homeomorphic to a trace of some knot, but is not diffeomorphic to any trace on any knot. I will use the trace embedding lemma to give examples of Stein not knot traces which are homotopy equivalent to the 2-sphere and admit topological locally flat spines but do not admit PL spines. Consequently, these 4-manifolds are simply connected but not geometrically simply connected. This gives a low-tech recovery of some recent work of Levine-Lidman and Kim-Ruberman. I will also use not knot traces to exhibit 3-manifolds Y that contain infinitely many knots in the same primitive homotopy class that are all concordant in some homology cobordism from Y to itself yet are not concordant in Y × I. Much of the work discussed in this lecture is joint with Kyle Hayden and Maggie Miller.