Geometric Topology

Abstract: We will discuss a recent result relating phenomena from two well-established, yet largely unrelated, subfields of 3-manifold topology. Specifically, we will demonstrate that the tightness of certain contact structures on a hyperbolic 3-manifold can be detected by the length and torsion of associated geodesics. No prior knowledge of contact topology or hyperbolic geometry is expected and we will give a brief introduction to both these fields. 

Abstract: Tight contact structures are the most interesting and studied object in contact geometry. Understanding how many distinct tight structures a given 3-dimensional manifold admits is quite a challenging and endearing problem.

In joint work with Hyunki Min, we classify tight contact structures on various surgeries on the Whitehead link, which provides the first classification result on an infinite family of hyperbolic L-spaces.

Abstract: A celebrated theorem of Ding and Geiges states that every connected, oriented, closed 3-manifold with a cooriented contact structure can be obtained via contact surgery along a Legendrian link in the standard tight contact 3-sphere. This leads to a complexity measure for contact 3-manifolds: the contact surgery number (csn), which is defined as the minimal number of components required in such a link presentation. The contact surgery numbers for the 3-sphere, certain lens spaces, and Brieskorn spheres have been studied extensively in the works of John Etnyre, Rima Chatterjee, Marc Kegel, and Sinem Onaran. 

In this talk, I will talk about the joint work with Marc Kegel which extends the study of csn to projective spaces by classifying all contact structures on 3-dimensional projective spaces that have contact surgery number one. 

Abstract: We introduce a generalization of Rasmussen's s-invariant, called the lasagna s-invariant, which assigns either an integer or -infinity to each second homology class of a smooth 4-manifold. The construction is based on the construction of Khovanov skein lasagna modules by Morrison-Walker-Wedrich. We present a few properties enjoyed by lasagna s-invariants, including some vanishing and nonvanishing results, and we show that they detect the exotic pair of knot traces X_{-1}(-5_2) and X_{-1}(P(3,-3,-8)), an example first discovered by Akbulut. This gives the first gauge/Floer-theory-free proof of the existence of compact orientable exotic 4-manifolds. This is joint work with Michael Willis.

We will discuss the following article.

Abstract: Given a proper embedding of an orientable surface in a 4-manifold, we study which surface diffeomorphisms can be induced by a diffeomorphism of the ambient 4-manifold. As an application, we answer a question on the existence of universal open books on a 5d manifold. We call a 5d open book universal if every 3d open book with connected binding admits an open book embedding in it.