Georgia Institute of Technology College of Sciences

How good of an invariant is the Jones polynomial? The question is closely tied to studying braid group representations since the Jones polynomial can be defined as a (normalized) trace of a braid group representation.

In this talk, I will present my work developing a new theory to precisely characterize the entries of classical braid group representations, which leads to a generic faithfulness result for the Burau representation of B_4 (the faithfulness is a longstanding question since the 1930s). In forthcoming work, I use this theory to furthermore explicitly characterize the Jones polynomial of all 3-braid closures and generic 4-braid closures. I will also describe my work which uses the class numbers of quadratic number fields to show that the Jones polynomial detects the unknot for 3-braid links - this work also answers (in a strong form) a question of Vaughan Jones.

I will discuss all of the relevant background from scratch and illustrate my techniques through simple examples.

https://gatech.zoom.us/my/margalit?pwd=b3RhY3pVZUdlRUR3S1FLZzhFR1RVUT09

There exist many different diagrammatic descriptions of 4-manifolds, with the usual claim that "such and such a diagram uniquely determines a smooth 4-manifold up to diffeomorphism". This raises higher order questions: Up to what diffeomorphism? If the same diagram is used to produce two different 4-manifolds, is there a diffeomorphism between them uniquely determined up to isotopy? Are such isotopies uniquely determined up to isotopies of isotopies? Such questions become important if one hopes to use "diagrams" to study spaces of diffeomorphisms between manifolds. One way to achieve these higher order versions of uniqueness is to ask that a diagram uniquely determine a contractible space of 4-manifolds (i.e. a 4-manifold bundle over a contractible space). I will explain why some standard types of diagrams do not do this and give at least one type of diagram that does do this.

An outstanding problem for surface bundles over surfaces, closely related to the symplectic geography problem in dimension four, is to determine for which fiber and base genera there are examples with non-zero signatures. I will report on our recent progress (joint with M. Korkmaz), which resolves the problem for all fiber and base genera except for 18 pairs at the time of writing.