Georgia Institute of Technology College of Sciences

Symmetric unions are an interesting class of knots. Although they have not been studied much for their own sake, they frequently appear in the literature. One such instance regards the question of whether there is a nontrivial knot with trivial Jones polynomial. In my talk, I will describe a class of symmetric unions, constructed by Tanaka, such that if any are amphichiral, they would have trivial Jones polynomial. Then I will show how such a knot not only answers the above question but also gives rise to a counterexample to the Cosmetic Surgery Conjecture.

In 1995, Mitsumatsu constructed a large family of Liouville domains whose topology obstructs the existence of a Weinstein structure.  Stabilizing these domains yields Liouville domains for which the topological obstruction is no longer in effect, and in 2019 Huang asked whether Mitsumatsu's Liouville domains were stably homotopic to Weinstein domains.  We answer this question in the affirmative.  This is joint work-in-progress with J. Breen.

Unlike the integral case, given a prime number p, not all Z/p-homology 3-spheres can be constructed as a Heegaard splitting with a gluing map an element of mod p Torelli group, M[p]. Nevertheless, letting p vary we can get any rational homology 3-sphere. This motivated us to study invariants of rational homology 3-spheres that comes from M[p]. In this talk we present an algebraic tool to construct invariants of rational homology 3-spheres from a family of 2-cocycles on M[p].

Artin groups are a generalization of braid groups, first defined by Tits in the 1960s. While specific types of Artin groups have many of the same properties as braid groups, other examples of Artin groups are still very mysterious. Braid groups are can be thought of as the mapping class groups of a punctured disc. The combinatorial and geometric structure of the mapping class group is reflected in a Gromov-hyperbolic space called the curve graph of the mapping class group.

A 3-manifold is called an L-space if its Heegaard Floer homology is "simple." No characterization of all such "simple" 3-manifolds is known. Manifolds obtained as the double-branched cover of alternating knots in the 3-sphere give examples of L-spaces. In this talk, I'll discuss the search for L-spaces among higher index branched cyclic covers of knots. In particular, I'll give new examples of knots whose branched cyclic covers are L-spaces for every index n. I will also discuss an application to "visibility" of certain periodic symmetries of a knot.

Ever since Eliashberg distinguished overtwisted from tight contact structures in dimension 3, there has been an ongoing project to determine which closed, oriented 3–manifolds support a tight contact structure, and on those that do, whether we can classify them. This thesis studies tight contact structures on an infinite family of hyperbolic L-spaces, which come from surgeries on the Whitehead link. We also present partial results on symplectic fillability on those manifolds.

One of interesting topic in low-dimensional topology is to study exotic smooth structures on closed 4-manifolds. In this talk, we will see an example to distinguish exotic smooth structure using H-slice knots.

In this talk I will discuss some new properties of an invariant for 4-manifold with boundary which was originally defined by Nobuo Iida. As one of the applications of this new invariant I will demonstrate how one can obstruct a knot from being h-slice (i.e bound a homologically trivial disk)  in 4-manifolds. Also, this invariant can be useful to detect exotic smooth structures of 4-manifolds. This a joint work with Nobuo Iida and Masaki Taniguchi.

Embedding problems, of an n-manifold into an m-manifold, can be heuristically thought to belong to a spectrum, from rigid, to flexible. Euclidean embeddings define the rigid end of the spectrum, meaning you can only translate or rotate an object into the target. Symplectic embeddings, depending on the object, and target, can show up anywhere on the spectrum, and it is this flexible vs rigid philosophy, and techniques developed to study them, that has lead to a lot of interesting mathematics.

I'm interested in the smooth mapping class group of S^4, i.e. pi_0(Diff^+(S^4)); we know very little about this group beyond the fact that it is abelian (proving that is a fun warm up exercise). We also know that every orientation preserving diffeomorphism of S^4 is pseudoisotopic to the identity (another fun exercise, starting with the fact that there are no exotic 5-spheres). Cerf theory studies the problem of turning pseudoisotopies into isotopies using parametrized Morse theory.