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. I’ll survey the literature on this subject. Then, I’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 ν 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.