We study braided embeddings, which is a natural generalization of closed braids in three dimensions. Braided embeddings give us an explicit way to construct lots of higher dimensional embeddings; and may turn out to be as instrumental in understanding higher dimensional embeddings as closed braids have been in understanding three and four dimensional topology. We will discuss two natural questions related to braided embeddings, the isotopy and lifting problem.
In 1941, Hopf gave a proof of the fact that the rational cohomology of a compact connected Lie group is isomorphic to the cohomology of a product of odd dimensional spheres. The proof is natural in the sense that instead of using the classification of Lie groups, it utilizes the extra algebraic structures, now known as Hopf algebras. In this talk, we will discuss the algebraic background and the proof of the theorem.
Continuing the theme of Hopf algebras, we will discuss a recipe by Reshetikhin and Turaev for link invariants using representations of quantum groups, which are non-commutative, non-cocommutative Hopf algebras. In the simplest case with the spin 1/2 representation of quantum sl2, we recover the Kauffman bracket and the Jones polynomial when combined with writhe. Time permitting, we will also talk about colored Jones polynomials and connections to 3-manifold invariants.
In the setting of manifolds with connected torus boundary, we can reinterpret bordered invariants as immersed curves in the once punctured torus. This machinery, due to Hanselman, Rasmussen, and Watson, is particularly useful in the context of knot complements. We will show how a type D structure can be viewed as a multicurve in the boundary of a manifold, and we will consider how the operation of cabling acts on this new invariant.
Bordered Floer homology, due to Lipshitz, Ozsváth, and Thurston, is a Heegaard Floer homology theory for 3-manifolds with connected boundary. This theory associates to the boundary surface (with suitable parameterization) a differential graded algebra A(Z). Our invariant comes in two versions: a left differential (type D) module over A(Z), or its dual, a right A-infinity (type A) module over A(Z).
An embedding of a manifold into a trivial disc bundle over another manifold is called braided if projection onto the first factor gives a branched cover. This notion generalizes closed braids in the solid torus, and gives an explicit way to construct many embeddings in higher dimensions. In this talk, we will discuss when a covering map of surfaces lift to a braided embedding.
In the world of 4 manifolds, finding exotic structures on 4 manifolds is considered one of most interesting and difficult problems. I will give a brief history of this and explain a very interesting tool "knot surgery" defined by Fintushel and Stern. In this talk I will mostly focused on drawing pictures. If time permits, I will talk various interesting applications.
A useful way of studying contact 3 manifolds is by looking at their open book decompositions. A result of Akbulut-Ozbagci, Ghiggini, and Loi-Piergallini showed that the manifold is filled by a Stein manifold if and only if the monodromy of an open book can be factorised as the product of positive Dehn twists. Then, the problem of classifying minimal fillings of contact 3 manifolds, or answering questions about which manifolds can be realised by Legendrian surgery, becomes questions about finding factorisations for a given mapping class.
In the first talk I will introduce the main constructions, many of which are classical, from scratch. This part will be introductory and accessible to a general audience with a basic knowledge of topology. This introduction will also serve as preparation for the main talk in which I will outline the proof and discuss some applications.
