Topology in dimension 3.5 Abstracts

$L^2$-acyclic bordism groups of 3-manifolds, Jae Choon Cha (POSTECH, Korea). Lecture Video

Recently Sylvain Cappell, Jim Davis and Shmuel Weinberger studied $L^2$-acyclic bordism groups of oriented closed manifolds in high dimensions. I will talk about the case of dimension three, introducing further invariants to study the structure peculiar to this dimension. In particular I will give answers to questions of Cappell, Davis and Weinberger about the 3-dimensional $L^2$-acyclic bordism group over the infinite cyclic group and the knot concordance group.

Every genus 1 algebraically slice knot is 1-solvable, Christopher Davis (University of Wisconsin at Eau Claire). Lecture Video

Joint with Taylor Martin, Carolyn Otto, and Jung Hwan Park. In the 1990's Cochran Orr and Teichner introduced a filtration of knot concordance indexed by half integers (the solvable filtration.) Since then this filtration has been a convenient setting for many advances in knot concordance. There are now many results in the literature demonstrating the difference between the n'th and (n.5)'th terms in this filtration, but none regarding the difference between the (n.5)'th and (n+1)'st. In this talk we will prove that every genus one (0.5)-solvable knot is 1-solvable. We will also provide a new sufficient condition for a high genus (0.5)-solvable knot to be 1-solvable and close with some possible candidates for knots which are (0.5)-solvable but not 1-solvable.

Noncommutative Alexander polynomials and the Thurston norm, Stefan Friedl (Universität Regensburg). Lecture Video

Tim Cochran and Shelly Harvey introduced noncommutative Alexander polynomials and showed that they give lower bounds on the Thurston norm. We show that given any 3-manifold there always exist noncommutative Alexander polynomials that detect the Thurston norm, and we will apply this to study the question of whether an epimorphism of knot groups implies an inequality of the corresponding genera.

Annular Khovanov-Lee homology and Braids, Elisenda Grigsby (Boston College). Lecture Video

Khovanov homology associates to a knot K in the three-sphere a bigraded vector space arising as the homology groups of an abstract chain complex. Using a deformation of Khovanov's complex, due to Lee, Rasmussen defined an integer-valued knot invariant he called s(K) that gives a lower bound on the 4-ball genus of knots, sharp for knots that can be realized as quasipositive braid closures. On the other hand, when K is a braid closure, its Khovanov complex can itself be realized in a natural way as a deformation of a triply-graded complex, defined by Asaeda-Przytycki-Sikora, further studied by L. Roberts, and now often referred to as the sutured annular Khovanov complex.

In this talk, I will describe joint work in progress with Tony Licata aimed at understanding an annular version of Lee's deformation of the Khovanov complex. In particular, we obtain a family of real-valued braid conjugacy class invariants generalizing Rasmussen's "s" invariant that give bounds on the Euler characteristic of smoothly-imbedded surfaces in the thickened solid torus. The algebraic model for this construction is the recently-defined Upsilon invariant of Ozsvath-Stipsicz-Szabo.

Knot theory and complex curves in subcritical Stein domains, Matthew Hedden (Michigan State University)

Beautiful results of Rudolph and Boileau-Orevkov characterize those links in the $3$-sphere which arise as transverse intersections with algebraic curves in $\mathbb{C}^2$. I'll discuss work in progress which provides a corresponding characterization of such links in connected sums of $S^1\times S^2$, viewed as the boundary of a subcritical Stein domain (the ball union Stein $1$-handles). Time permitting, I'll talk about further generalizations being pursued with Baykur, Etnyre, Kawamuro, and Van Horn-Morris.

Knot Floer homology and concordance, Jennifer Hom (Georgia Institute of Technology). Lecture Video

We will discuss applications of knot Floer homology to concordance. In particular, if two knots are concordant, then their knot Floer complexes satisfy a certain type of stable equivalence. We will relate the notion of stable equivalence to several other concordance invariants coming from the knot Floer complex.

The generalized doubling operad, Constance Leidy (Wesleyan University). Lecture Video

For decades, much of the focus of the study of knots up to concordance has been on the structure of this as an abelian group. Recent work has shown that this perspective may be too limited. We will describe some alternative structures on this set. In particular, we will consider structure that results from a new "generalized doubling operad".

Satellite operators and piecewise-linear concordance, Adam Levine (Princeton University)

Every knot in the 3-sphere bounds a piecewise-linear (PL) disk in the 4-ball, but Akbulut showed in 1990 that the same is not true for knots in the boundary of an arbitrary contractible 4-manifold. We strengthen this result by showing that there exists a knot K in a homology sphere Y (which is the boundary of a contractible 4-manifold) such that K does not bound a PL disk in any homology 4-ball bounded by Y. The proof relies on using bordered Heegaard Floer homology to show that the action of a certain satellite operator on the knot concordance group is not surjective.

Exploring Tim's legacy, a slice at a time, Kent Orr (Indiana University). Lecture Video

Tim painted a broad mathematical canvas, working on high dimensional knots and links, the transfinite lower central series of 3-manifold groups, finite type invariants 3-manifolds, 4-dimensional topological surgery theory, linking forms on 3-manifolds, the derived filtration on the classical knot concordance group along with L²-methods and von-Neumannρ-invariants developed to study filtrations, and smoothing theory in dimension four. Tim's work motivated and fundamentally underpinned future directions in his field.

I will focus on a small number of Tim's most substantial and influential papers, emphasizing the state of the subject both before and after Tim's contributions, and how his work redirected the subject.

Milnor's triple linking number and derivatives of knots, Jung Hwan Park (Rice University). Lecture Video

A derivative of an algebraically slice knot K is an oriented link disjointly embedded in a Seifert surface of K such that its homology class forms a basis for a metabolizer of K. We use a ribbon obstruction related to Milnor's triple linking number. As an application we disprove the n-solvable filtration version of Kauffman's conjecture assuming that 0.5-solvable knot is also 1-solvable. In addition, for some algebraically slice knots with a fixed metabolizer, we get a complete understanding of Milnor's triple linking number of derivatives associated to the metabolizer. This is joint work with Mark Powell.

Gropes and metrics on the knot concordance set, Mark Powell (Université du Québec à Montréal). Lecture Video

It was posited by Cochran, Harvey and Leidy that knot concordance ought to exhibit some kind of fractal structure. A grope is a special type of 2-complex built as a union of surfaces with boundary, that approximates a disc. We will associate a real number to a grope, that measures its failure to be a disc. By considering embeddings of these objects in 4-space, we will define a family of pseudo-metrics on the set of concordance classes of knots. In the talk, which is on joint work with Tim Cochran and Shelly Harvey, I will define these notions, and I will discuss the interesting properties that our metrics possess. In particular, there are sequences of knots whose distance from the unknot stays positive but tends to zero.

Satellite operations and fractal structures on knot concordance, Arunima Ray (Brandeis University). Lecture Video

Satellite operations are a natural generalization of the connected sum operation on knots. I will discuss a number of recent results about satellite operations on topological and smooth knot concordance classes, particularly winding number one satellites, contributing to the conjecture that the concordance groups are fractal spaces. Some of the results are joint work with Tim Cochran and/or Christopher Davis.

Configurations of embedded spheres, Daniel Ruberman (Brandeis University)

Configurations of lines in the plane have been studied since antiquity. In recent years, combinatorial methods have been used to decide if a specified incidence relation between certain objects ("lines") and other objects ("points") can be realized by actual points and lines in a projective plane over a field. For the real and complex fields, one can weaken the condition to look for topologically embedded lines (circles in the real case, spheres in the complex case) that meet according to a specified incidence relation.

I will explain some joint work with Laura Starkston (Stanford) giving new topological restrictions on the realization of configurations of spheres in the complex projective plane, and some new constructions of configurations that can be realized smoothly.

Cochran’s $\beta^i$-invariants via twisted Whitney towers, Peter Teichner (Max Planck Institute for Mathematics and University of California, Berkeley). Lecture Video

In joint work with Jim Conant and Rob Schneiderman, we showed that Tim Cochran's invariants $\beta^i(L)$ of a $2$-component link $L$ in the $3$--sphere can be computed as intersection invariants of certain 2-complexes in the $4$--ball with boundary $L$. We’ll explain these 2-complexes as special types of twisted Whitney towers, which we call {\em Cochran towers}, and which exhibit a new phenomenon: A Cochran tower of order $2k$ allows the computation of the $\beta^i$ invariants for all $i\leq k$, i.e.\ simultaneous extraction of invariants from a Whitney tower at multiple orders, in contrast with the order $n$ Milnor invariants which require an order $n$ Whitney tower.