# Seminar on knot concordance and 4-manifolds

**Venue:** HIM lecture hall, Poppelsdorfer Allee 45, Bonn (unless stated otherwise)

Organizers: Christopher Davis, Mark Powell

## Friday, September 9

10:00 - 12:00 Peter Teichner (MPIM, Bonn): Introduction to Whitney towers

## Friday, September 16

16:30 - 17:30 Arunima Ray (Brandeis University): 4-dimensional analogues of Dehn's lemma

## Monday, September 19

10:00 - 12:00 Peter Teichner (MPIM, Bonn): Twisted Whitney towers and Milnor's invariants

## Friday, September 23

16:30 - 17:30 Jung Hwan Park (Rice University): Milnor's triple linking number and derivatives of knots

Abstract: 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.

## Wednesday, September 28

11:00 - 12:00 Stefan Friedl (Universität Regensburg): How ??? to define a link concordance invariant

## Friday, October 7

13:30 - 14:30 Duncan McCoy (University of Glasgow): On L-space knots obtained from unknotting arcs in alternating diagrams

## Wednesday, October 12

10:30 - 11:30 Juanita Pinzón-Caicedo (University of Georgia): Examples of relative trisections

Abstract: Trisections of 4-manifolds relative to their boundary were introduced by Gay and Kirby in 2012. They are decompositions of 4-manifolds that induce open book decomposition in the bounding 3-manifolds. This talk will focus on diagrams of relative trisections and will be divided in two. In the first half I will focus on trisections as fillings of open book decompositions and I will present different fillings of different open book decompositions of the Poincare homology sphere. In the second half I will show examples of trisections of pieces of some of the surgery techniques that result in exotic 4-manifolds.

## Friday, October 14

15:00 - 16:00 Miriam Kuzbary (Rice University): Knotified link concordance

16:30 - 17:30 Alexander Zupan (University of Nebraska-Lincoln): The Generalized Property R Conjecture and fibered ribbon knots

Abstract: The Generalized Property R Conjecture (GPRC, Kirby Problem 1.82) proposes a characterization of n-component links in the 3-sphere with a Dehn surgery to the connected sum of n copies of S^{1} x S^{2}. The case n=1 is classical Property R, proved by Gabai in 1987. However, in 2010, Gompf, Scharlemann, and Thompson proposed a family of potential counterexamples in the case n=2. We use fibered ribbon knots to generalize the construction of Gompf, Scharlemann, and Thompson, giving potential counterexamples to the GPRC for all even n. This is joint work with Jeffrey Meier.

## Monday, October 24

13:30 - 14:30 Shelly Harvey (Rice University): Cut number of homology handlebodies

## Tuesday, October 25

Venue: HIM seminar room (in the basement), Poppelsdorfer Allee 45

15:00 - 16:00 Kyungbae Park (KIAS): ℤ^{∞}-summands of knots with Alexander polynomial 1

## Wednesday, October 26

15:00 - 16:00 Paolo Aceto (Renyi Institute of Mathematics): Knot concordance and rational homology cobordism

16:30 - 17:30 Matt Hedden (Michigan State University): Knot Floer homology and generalized adjunction inequalitites

## Thursday, October 27

13:00 - 14:30 Lukas Lewark (Universität Bern): Slice genus bounds from the Seifert form

## Friday, November 4

16:30 - 17:30 Maciej Borodzik (University of Warsaw): Involutive Floer homology and rational cuspidal curves

Abstract: Using Involutive Floer theory of Hendricks and Manolescu we show a new criterion obstructing the existence of rational cuspidal curves with some configurations of singular points. The criterion is especially effective for rational cuspidal curves with two singular points. This is a joint project with Jen Hom.

## Tuesday, November 8

16:30 - 17:30 Stefan Behrens (Universiteit Utrecht): Seiberg-Witten theory and homotopy theory (part 1)

Abstract: The Seiberg-Witten equations were introduced by Witten in the mid '90s. Ever since then, they have played in important role in the study of 3- and 4-dimensional manifolds. A recent highlight of the 3-dimensional theory is Manolescu's resolution of the triangulation conjecture. The goal of these lectures is to give an overview of the homotopy theoretic approach to Seiberg-Witten theory that was pioneered by Bauer, Furuta, and Manolescu. I will give a glimpse at the inner workings of this machine and indicate as many applications as time permits.

## Wednesday, November 9

12:20 - 13:20 Sander Kupers (Københavns Universitet): Finiteness properties of automorphisms of manifolds

Abstract: Combining several recent results in the theory of manifolds, we will prove that the homotopy groups o diffeomorphisms of an n-dimensional disk (n ≠ 4, 5, 7) are finitely generated.

16:30 - 17:30 Stefan Behrens (Universiteit Utrecht): Seiberg-Witten theory and homotopy theory (part 2)

## Thursday, November 10

16:30 - 17:30 Stefan Behrens (Universiteit Utrecht): Seiberg-Witten theory and homotopy theory (part 3)

## Wednesday, November 16

16:30 - 17:30 Marco Golla (Uppsala University): Signature defects, handles, and ribbon discs

Abstract: The homology groups of a manifold give a lower bound on the number of handles in a handle decomposition (or even on the cells of a CW decomposition). We use Casson-Gordon signatures to improve on this bound for rational homology 4-balls bounding a given rational homology 3-sphere. In turn, this gives information about slice and ribbon discs for knots in the 3-sphere. This is joint work with Paolo Aceto and Ana Lecuona.

## Friday, November 25

16:30 - 17:30 Min Hoon Kim (KIAS): Ideal classes and Cappell-Shaneson homotopy 4-sphere

Abstract: Gompf proposed a conjecture on Cappell-Shaneson matrices whose affirmative answer implies that all Cappell-Shaneson homotopy 4-spheres are diffeomorphic to S^{4}. We study Gompf conjecture on Cappell-Shaneson matrices using Aitchison and Rubinstein's number theoretic approach based on Latimer-MacDuffee theorem. We find a hidden symmetry between trace n Cappell-Shaneson matrices and trace 5-n Cappell-Shaneson matrices which was suggested by Gompf experimentally. Hinges on this symmetry, we prove that Gompf conjecture for the trace n case is equivalent to the trace 5-n case. We confirm Gompf conjecture for the special cases that -69 ≤ trace ≤ 74 and trace ≠ -68, -65, 70, 73 and corresponding Cappell-Shaneson homotopy 4-spheres are diffeomorphic to S^{4}. This is joint work with Shohei Yamada.

## Wednesday, November 30

16:30 - 17:30 Jonathan Hillman (The University of Sydney): Solvable 2-knot groups

## Thursday, December 1

15:00 - 16:00 Arunima Ray (Brandeis University): Knots in dimension 3.5

This is an informal talk.

## Thursday, December 15

16:30 - 17:30 Ana Lecuona (Aix-Marseille Université): Splice links and coloured signatures

Abstract: The splice of two links is an operation defined by Eisenbund and Neumann that generalizes several other operations on links, such as the connected sum, cabling or the disjoint union. There has been much interest in understanding the behaviour of different link invariants under the splice operation (genus, whether they fibre, Conway polynomial, Heegaard-Floer homology among others) and the goal of this talk is to present a formula relating the coloured signature of the splice of two oriented links to the coloured signatures of its two constituent links. As an immediate consequence, we have that the conventional univariate Levine-Tristram signature of a splice depends, in general, on the coloured (or multivariate) signatures of the summands. This is a joint work with Alex Degtyarev and Vincent Florens.