Ver. 23, Mar., 2017

Workshop Diﬀerential Topology
The Univ. of Electro-Comm（Room: NewC403）1-5-1 Chofu-ga-oka, Chofu, Tokyo
28 – 31, March, 20173/28 (Preworkshop for graduates (in Japanese))
3/29-31 (Workshop for experts (in English))Subject：Heegaard Floer homology and its related topics
Keywords : Heegaard Floer homology, L-space, contact structure, concordance invariant
Schedule
10:00–11:00
11:20–12:20
14:20–15:20
15:40–16:40
17:00–18:00

28 Tue.
M. Tange
Y. Bao29 Wed.
K. Ichihara
H. Matsuda

30 Thu.
K. Park
Z. Wu

31 Fri.
D. Krcatovich
K. SatoLunch

Lunch

Lunch

Lunch

K. SatoY. Nakae
T. AbeY. BaoZ. Wu
M. Kim

T. ObaK. Motegi
T. Ito28 Tuesday
10:00–11:00 Motoo Tange (University of Tsukuba)
Introduction to Heegaard Floer homology
11:20–11:20 Yuanyuan Bao (The University of Tokyo)
The Heegaard Floer homology for a knots, link and 3-manifold with boundary
14:20–15:20 Kouki Sato (Tokyo Institute of Technology)
A survey of knot concordance invariants from Heegaard Floer homology
15:40–16:40 Yasuharu Nakae (Akita University)
Taut foliations, its properties and constructions
17:00–18:00 Tetsuya Abe (Osaka City University)
Knot Floer homology and the enhancement to the Milnor number

29 Wednesday
10:00–11:00 Kazuhiro Ichihara (Nihon University)
Non-leftordarable surgeries on twisted torus knots
11:20–11:20 Hiroshi Matsuda (Yamagata University)
Homological invariants of surface-knots

14:20–15:20 Yuanyuan Bao (The University of Tokyo)
The Alexander polynomial of the balanced bipartite graph
15:40–16:40 Zhongtao Wu (The Chinese University of Hong Kong)
Some minimal genus problems in low dimensional topology
17:00–18:00 Min Hoon Kim (KIAS, Korea)
On the bipolar filtration of topologically slice knots
18:30–Banquet En’ya at Chofu North

30 Thursday
10:00–11:00 Kyungbae Park (KIAS, Korea)
Surgery obstructions and Heegaard Floer theory
11:20–11:20 Zhongtao Wu (The Chinese University of Hong Kong)
Some minimal genus problems in low dimensional topology
14:20–15:20 Takahiro Oba (Tokyo Institute of Technology)
Survey of Heegaard Floer homologies and contact structures
15:40–16:40 Kimihiko Motegi (Nihon University)
Twistings, band-sums and L-space surgeries
17:00–18:00 Tetsuya Ito (Osaka University)
L-space and left-ordering

31 Friday
10:00–11:00 David Krcatovich (Rice University)
Cobordisms of knots, braid index, and the Upsilon invariant
11:20–11:20 Kouki Sato (Tokyo Institute of Technology)
A full-twist inequality for the nu+ invariant
the End of the Program

Organizers：
Tetsuya ABE (Osaka City University),
Motoo TANGE (University of Tsukuba),
Yuichi YAMADA (The Univ. of Electro-Comm.)
Supported by
JSPS, Grant-in-Aid for Scientific Research (C) 16K0514,
JSPS, Grant-in-Aid for Young Scientists (B) 16K17597,
JSPS, Grant-in-Aid for Young Scientists (B) 26800031

Diﬀerential Topology 17

1Workshop : Diﬀerential Topology 17
Abstracts
Motoo Tange (University of Tsukuba)
Introduction to Heegaard Floer homology
Abstract: In this lecture we begin with the definition of Heegaard Floer homology
for any closed 3-manifold and includes some invariants and application. We also givea definition of corretion term and several computations and applications, for example,
plumbed 3-manifolds and Dehn surgeries of a knot.

Yuanyuan Bao (The University of Tokyo)
The Heegaard Floer homology for a knots, link and 3-manifold with boundary
Abstract: On the presupposition that Heegaard Floer homology (HF) for a closed 3manifold has been introduced, in this talk, we will review the definition of the HF for alink and knot and its properties. This definition turns out to be a special case of the HF
for a 3-manifold with boundary. If time allows, we will briefly introduce the definitionof the HF for a 3-manifold with boundary (mainly sutured Floer homology and bordered
Floer homology).

Kouki Sato (Tokyo Institute of Technology)
A survey of knot concordance invariants from Heegaard Floer homology
Abstract: In this talk, we discuss several knot concordance invariants derived from Heegaard Floer homology. In particular, we focus on the correction terms of Dehn surgeries,
Ni-Wu’s Vk invariants and Hom-Wu’s ν + invariant.

Yasuharu Nakae (Akita University)
Taut foliations, its properties and constructions
Abstract: Let F be a codimension one foliation of a closed 3-manifold M . F is called
a taut foliation if all leaves of F have a closed transversal. By the definition, F has noReeb component. Then a taut foliation F is Reebless and it has some properties thatthe fundamental group of M is infinite, M is irreducible and the universal cover of M
is homeomorphic to 3-dimensional Euclid space by the theorems of Novikov, Rosenberg
and Palmeira. I will explain these properties and some example of construction of tautfoliations on 3-manifolds, especially focus on a knot complement. If there is a time, I will
explain a relation between an L-space and existence of taut foliations.

Diﬀerential Topology 172Tetsuya Abe (Osaka City University)
Knot Floer homology and the enhancement to the Milnor number
Abstract: First, I will explain basic facts on fibered links and contact structures. Secondly, I will explain the relation between the enhancement to the Milnor number of fiberedknots (or the d3 -invariant of fibered knots) and the Knot Floer homology of fibered knots.

Kazuhiro Ichihara (Nihon University)
Non-leftordarable surgeries on twisted torus knots
Abstract: The well-known L-space conjecture saids that an irreducible rational homology 3-sphere is an L-space if and only if its fundamental group is not left-orderable. One ofthe known approaches to the conjecture is by using Dehn surgery. In this talk, I will talkabout Dehn surgeries on twisted torus knots yielding 3-manifolds with non-leftorderable
fundamental groups. This talk is based on joint works with Yuki Temma and Yasuharu
Nakae.

Hiroshi Matsuda (Yamagata University)
Homological invariants of surface-knots
Abstract:

Yuanyuan Bao (The University of Tokyo)
The Alexander polynomial of the balanced bipartite graph
Abstract: The Alexander polynomial is one of the most classical knot invariants. It has
both classical and modern interpretations. In classical case for example, it can be definedfrom the universal abelian covering of the knot complement, while in modern case, it has
super sl(1, 1) model, Burau representation model etc. For a balanced bipartite graph, wedefined its Heegaard Floer homology, and we call its Euler characteristic the Alexanderpolynomial of the graph. In this talk, we aim to discuss diﬀerent interpretations of thispolynomial.

Zhongtao Wu (The Chinese University of Hong Kong)
Some minimal genus problems in low dimensional topology
Abstract: I plan to give a series of talk on minimal genus problems in low dimensionaltopology. we will discuss the notion of the rational genus of a knot and focus on theminimal genus problem for knots in a rational homology three-sphere that represent agiven homology class. This is joint work with Yi Ni.

Min Hoon Kim (KIAS, Korea)

Diﬀerential Topology 17

3On the bipolar filtration of topologically slice knots
Abstract: Celebrated theorems of Freedman and Donaldson have an immediate corollarythat there is a topologically slice knot which is not smoothly slice. Let T be the subgroupof the smooth knot concordance group of topologically slice knots. Understanding thestructure of T is of fundamental importance since T measures the subtle diﬀerence between
topological and smooth category in dimension 4. Cochran, Harvey and Horn proposed abeautiful framework to study T systematically by introducing a geometrically defined
filtration on T which is called the bipolar filtration. Cochran, Harvey and Horn could
interpret many knot concordance invariants in terms of the bipolar filtration. Up to now,the non-triviality of this filtration was settled only at the zeroth level and the first level.
In this talk, we prove that the bipolar filtration on T is highly non-trivial at every level.
The proof involves both Cheeger-Gromov L2 ρ-invariants and Heegaard Floer d-invariants.
This is joint work with Professor Jae Choon Cha.

Kyungbae Park (KIAS, Korea)
Surgery obstructions and Heegaard Floer theory
Abstract: It is well known that any closed oriented 3-manifold can be obtained bysurgery on a link in the 3-sphere. It is hard in general to answer: Which 3-manifolds can,or cannot, be obtained by surgery on a knot (single component link)? In this talk, we willsurvey results related to this question, responded by using Heegaard Floer theory. We
also introduce our result on 3-manifolds with the first betti-number 1. This is a joint workwith Matt Hedden and Min Hoon Kim.

Zhongtao Wu (The Chinese University of Hong Kong)
Some minimal genus problems in low dimensional topology
Abstract: I plan to give a series of talk on minimal genus problems in low dimensional
topology. we will discuss the notion of the rational genus of a knot and focus on theminimal genus problem for knots in a rational homology three-sphere that represent agiven homology class. This is joint work with Yi Ni.

Takahiro Oba (Tokyo Institute of Technology)
Survey of Heegaard Floer homologies and contact structures
Abstract: Ozsbáth and Szabó introduced an invariant for a given contact 3-manifold
by using Heegaard Floer homologies. In this talk, first we recall the definition of thisinvariant, and then we review some properties and results of this invariant. Particular, weexhibit an alternative description of this invariant given by Honda, Kazez and Matic.

Kimihiko Motegi (Nihon University)

Diﬀerential Topology 17

4Twistings, band-sums and L-space surgeries
Abstract: Twisting operations and band-sum operations enable us to create new knots
form given ones.We will discuss if we can obtain L-space knots using these operations. We
give a necessary and suﬃcient condition for a twist family of knots to contain infinitely
many L-space knots. We also prove that a non-trivial band sum never yields an L-spaceknot. This generalizes Krcatovich’s result which asserts that a non-trivial connected sum
never yields an L-space knot.
This is joint work with Ken Baker.

Tetsuya Ito (Osaka University)
L-space and left-ordering
Abstract: It is conjectured that the fundamental group of 3-manifold is non-left-orderable
if and only if it is L-space. In this talk I will review basic of left-ordering on group andhow it is (expected to be) related to topology and geometry of 3-manifolds.

David Krcatovich (Rice University)）
Cobordisms of knots, braid index, and the Upsilon invariant
Abstract: If two knots occur as intersections of a given algebraic curve in C 2 withconcentric three-spheres, they are connected by a cobordism whose genus is the diﬀerenceof the slice genera of the two knots. A cobordism with this genus is called optimal. We
use the Upsilon invariant of Ozsvath-Stipsicz-Szabo to obstruct the existence of optimalcobordisms. In doing so, we generalize a result of Morton-Franks-Williams on the minimal
braid index of knots with a ”full twist”, and show that in some cases Upsilon can provide
a lower bound on the braid index of any representative in a concordance class of knots.
This is joint work with Peter Feller.

Kouki Sato (Tokyo Institute of Technology)
A full-twist inequality for the nu+ invariant
Abstract: Hom and Wu introduced a knot concordance invariant called nu+ (ν + ), which
dominates many concordance invariants derived from Heegaard Floer homology. In thistalk, we give a full-twist inequality for nu+. By using the inequality, we extend Wu’s
cabling formula for ν + (which is proved only for particular positive cables) to all cablesin the form of an inequality. In addition, we also discuss ν + -equivalence, which is anequivalence relation on the knot concordance group. We introduce a partial order on
ν + -equivalence classes, and study its relationship to full-twists.
The End