## 2018 Mini-workshop on Low Dimensional Topology

September 14-15, 2018
POSTECH, Pohang, South Korea

(Click photo for a full-resolution version)

### Invited Speakers

BoGwang Jeon (POSTECH)

Min Hoon Kim (KIAS)

Se-Goo Kim (Kyung Hee University)

Taehee Kim (Konkuk University)

Motoo Tange (University of Tsukuba)

A joint event of this mini-workshop is a special colloquium of IBS-CGP and Department of Mathematics, on September 14, 5:00pm:

Speaker: Peter Teichner (Max Planck Institute for Mathematics)
Title: Topological phases, field theories and manifold invariants

### Time table (tentative)

All lectures are given at Math Building 404.

September 14, 2018

 09:40am-10:40am A family of freely slice good boundary links Min Hoon Kim Abstract. The still open topological surgery conjecture for 4-manifolds is equivalent to the statement that all good boundary links are freely slice. In this talk, I will show that every good boundary link with a pair of derivative links on a Seifert surface satisfying a homotopically trivial plus assumption is freely slice. This subsumes all previously known methods for freely slicing good boundary links with two or more components, and provides new freely slice links. This is joint work with Jae Choon Cha and Mark Powell. 11:00am-12:00pm Volumes of hyperbolic 3-manifolds BoGwang Jeon Abstract. The following is a well-known theorem of Thurston: For a 1-cusped hyperbolic 3-manifold $M$, there are only finitely many hyperbolic Dehn fillings of M having the same volume. In this talk, I will present my strategy to prove the following theorem: For a 1-cusped hyperbolic 3-manifold $M$, there exists $c >0$ such that the number of hyperbolic Dehn fillings of M having the same volume is less than $c$. 02:30pm-03:30pm Integer valued knot concordance invariants Se-Goo Kim Abstract. There are many known integer valued knot concordance invariants, such as Rasmussen s-invariant and Ozsváth-Szabó tau-invariant, whose values coincide with knot signatures for alternating knots, after normalized. We introduce some local moves for an almost alternating knot having the same value on these invariants to produce another almost alternating knot having the same value on these invariants. 05:00pm-06:00pm Topological phases, field theories and manifold invariants Peter Teichner’s Special Math Colloquium Abstract. After recalling the Atiyah-Segal-Witten formalism for topological field theories (TFTs), we will explain some recent computations of Freed-Hopkins in the case of positive invertible TFTs. Their tables magically agree with computations made in condensed matter physics of gapped systems, namely for symmetry protected topological phases. In both approaches, the input is the space-time dimension $d$, together with a symmetry group H, and the output is a finitely generated abelian group $TP(d,H)$ of topological phases. It remains an open question why these groups can be computed in two completely different ways. For fixed dimension $d$, there is a 10-fold way in which the groups H arise, and we’ll show how they are related to the $8+2$ super division algebras (over the real and complex numbers). We’ll prove that invertible TFTs are classified by their partition function, an invariant of closed $d$-manifolds with structure group H. Finally, we’ll characterize such manifold invariants in terms of a 4-term cut-and-paste relation and connect these back to the Freed-Hopkins computations. The last part is current joint work with Matthias Kreck and Stephan Stolz.

#### September 15, 2018

 09:40am-10:40am Ribbon disk diagram in handle decomposition of B4  Motoo Tange Abstract. Any ribbon disk in $S^3$ is an immersion of a disk with arc-type singularity with ribbon condition. Our motivation of this study is to generalize the presentation of the disk with singularity to any slice disk condition. We show that moving the presentation of disk with singularity by using handle moves, we deform it into presentation with modified canceling pairs. 11:00am-12:00am The 4-genus of knots and links Taehee Kim Abstract. It is well known that a knot bounds an oriented compact connected surface in the 3-sphere, and hence in the 4-ball. The 4-genus of a knot is the minimal genus of all such surfaces in the 4-ball, and the 4-genus of a link is defined similarly. In this talk, using Cheeger-Gromov-von Neumann rho-invariants, I will present new examples of knots and links regarding the stable 4-genus and the filtrations on link concordance of Cochran-Orr-Teichner. This is joint work with Jae Choon Cha and Min Hoon Kim.

### Organizers

Jae Choon Cha (POSTECH) and Jihun Park (IBS-CGP/POSTECH)
If you have any inquiry, then please email to jccha@postech.ac.kr.

### Website

https://gt.postech.ac.kr/~jccha/2018-mini-workshop/