Georgia Institute of Technology College of Sciences

In this pair of talks I will survey some of the machinery developed by Conant, Schneiderman, and Teichner to study Whitney towers, and their applications to the study of knot and link concordance. Whitney towers can be thought of as measuring the failure of the Whitney trick in dimension 4 and can be used, in a sense, to approximate slice disks. The talks will be based on various papers of Schneiderman, Conant-Schneiderman-Teichner, Cochran-Orr-Teichner and lecture notes by those authors.

How do we build a knot table? We will discuss Conway’s paper “an enumeration of knots and links” and Hoste, Thistlethwaite and Weeks’ paper “the first 1701936 knots”.

In this talk I will briefly describe link Floer homology toolbox and its usefulness. Then I will show how link Floer homology can detect links with small ranks, using a rank bound for fibered links by generalizing an existing result for knots. I will also show that stronger detection results can be obtained as the knot Floer homology can be shown to detect T(2,8) and T(2,10), and that link Floer homology detects (2,2n)-cables of trefoil and figure eight knot. This talk is based on a joint work with Fraser Binns (Boston College).

A link L in the 3-sphere is called chi-slice if it bounds a properly embedded surface F in the 4-ball with Euler characteristic 1. If L is a knot, then this definition coincides with the usual definition of sliceness. One feature of such a link L is that if the determinant of L is nonzero, then the double cover of the 3-sphere branched over L bounds a rational homology ball. In this talk, we will explore the chi-sliceness of 3-braid links.

A well known result of Fox and Milnor states that the Alexander polynomial of slice knots factors as f(t)f(t^{-1}), providing us with a useful obstruction to a knot being slice. In 1978 Kawauchi demonstrated this condition for the multivariable Alexander polynomial of slice links.  In this talk, we will present an extension of this result for the multivariable Alexander polynomial of 1-solvable links. (Note: This talk will be in person) 

Links of singularity and generalized algebraic links are ways of constructing three-manifolds and smooth links inside them from algebraic surfaces and curves inside them. Némethi created lattice homology as an invariant for links of normal surface singularities which developed out of computations for Heegaard Floer homology. Later Ozsváth, Stipsicz, and Szabó defined knot lattice homology for generalized algebraic knots in rational homology spheres, which is known to play a similar role to knot Floer homology and is known to compute knot Floer in some cases.

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.