# トポロジーセミナー

## Seminar on Geometric Topology of dimension 3

#### 2018年度

アブストラクト：
For every genus g(≥ 3) Heegaard splitting of S^3, we give an example of two intersecting primitive disks D and E such that any disk surgery of D (resp. E) along an outermost disk in E (resp. D) does not yield primitive disks. This is a joint work with Sangbum Cho and Yuya Koda.

#### 2016年度

アブストラクト：
Agol は、円周上の擬アノソフモノドロミーを持つ穴あき曲面束に対して、そのモノドロミーの安定、不安定葉層構造の特異点をファイバー曲面から除いておけば、その曲面束は “標準的な’’ veering という性質を持った理想三角形分割を許容することを示した。一方、Epstein-Penner は、任意の有限体積カスプ付き完備双曲多様体は “標準的な’’ 幾何的理想多面体分割を許容することを示した。これらは例えば円周上の一点穴あきトーラス束では一致することが知られている。
この講演ではまず、双曲的ファイバー二橋絡み目補空間の Epstein-Penner による分割が veering になるかどうかの必要十分条件を与える。さらに veering であるような双曲的ファイバー二橋絡み目補空間の Epstein-Penner の分割に、補空間が曲面束であることを用いた組合せ的記述を与える。

アブストラクト：
Calegari-Sun-Wang は曲面の写像類に有限被覆とべきにより定義される通約性を定義した．

#### 2015年度

Date: 2015年8月6日（木）16:20～17:50

Place: 日本大学文理学部，3号館4階 3404教室

Speaker: Johann Andreas Makowsky (Department of Computer Science, Technion–Israel Institute of Technology)

Title: Why is the chromatic polynomial a polynomial?

Abstract: We give a proof different from Birkhoff’s proof of the fact that counting k-colorings of a graph gives rise to a polynomial in k, the chromatic polynomial of the graph. We use this to show that many counting functions on graphs are polynomials. We show that every univariate graph polynomial definable in Second Order Logic is a generalized chromatic polynomial. (Joint work with B. Zilber and T. Kotek)

この講演に先立ち同日の 14:40 〜 16:10 に同じ教室にて，

よろしければ、こちらにも、ご参加ください。

Abstract: In this talk, we explain the relationship between open book decompositions of 3-manifolds and their supported contact structures. The idea was given by Thurston and Winkelnkemper and developed by Giroux. We first introduce basic facts in 3-dimensional contact topology as Darboux theorem, Gray stability and Lutz twists and then explain Giroux’s results on convex surfaces and open book decompositions.

Abstract: We show that a random link via random braid model is a hyperbolic link $L$, moreover,
every strictly non-trivial Dehn filling on $S^{3}- L$ is a hyperbolic closed 3-manifold.

#### 2014年度

Abstract:
We consider a random walk on the mapping class group of a surface of finite type. We assume that the random walk is determined by a probability measure whose support is finite and generates a non-elementary subgroup H. We further assume that H is not consisting only of lifts with respect to any one covering. Then we prove that the probability that the random walk gives a non-minimal mapping class in its fibered commensurability class decays exponentially. As an application of the minimality, we prove that for the case where a surface has at least one puncture, the probability that the random walk gives a mapping class with an arithmetic mapping torus decays exponentially.

Abstract:
Knots in handlebodies with boundary reducing surgeries were thought to have small bridge number. We show that this is not the case by constructing a family of knots with nontrivial surgeries yielding handlebodies. The construction is given by “twisting along an annulus,” and using a result of Baker-Gordon-Luecke, we show that there are knots in the family with arbitrarily large bridge number. This work is joint with Ken Baker and John Luecke.

Abstract : 今回お話しすることは門上晃久氏（華東師範大学）と円山憲子氏（武蔵野美術大学）との共同研究の結果です。

#### 2013年度

Abstract : Manifolds are commensurable if they have a common cover, of finite degree over each. If two hyperbolic manifolds are commensurable, they have the same invariant trace field, Q-dependent volume, Q-dependent Bloch invariants and PGL2(Q)-dependent cusp parameters. In [1], E. Chesebro and J. Deblois has constructed a family of hyperbolic link complements which have the same volume, trace field, Q-dependent Bloch invariants and PGL2(Q)-dependent cusp parameters. But they did not show the incommensurability of these link complements. In this talk, we show that these are incommensurable each other.

[1] E. Chesebro and J. Deblois Algebraic invariants, mutation, and commensurability of link complements, Preprint (2012) arXiv:1202.0765

Abstract :We show that there is an infinite family of knots with an unexpected property. In fact, for each knot k in the family there is an incompressible branched surface B properly embbeded in the knot exterior, which carries Klein bottles and tori with an arbitrarily large number of boundary components. This is joint work with Enrique Ramirez-Losada.

Abstract :This is an introductory talk to our work about the distance of Heegaard splittings of hyperbolic 3-manifolds.
A (complete, finite volume) hyperbolic 3-manifold is a 3-manifold equipped with a riemannian metric with certain very nice properties. The volume of a hyperbolic 3-manifold M, denoted by Vol(M), is a measure of its complexity.

A Heegaard surface for a 3-manifod M is a surface F embedded in M that decompose M into two simple pieces called handlebodies. There are two ways in which we say that F is simple: first, if it genus is small. Second, if its distance (as defined by Hempel using the curve complex) is at most 2. (We will explain this condition in the talk.)

Our goal is to show that a Heegaard surface F of a simple hyperbolic 3-manifold is itself simple; more precisely we prove:

THEOREM: There exists a constant L>0 so that if M is a generic hyperbolic 3-manifold and F is a Heegaard surface for M, then either g(F) < LVol(M), or the distance of F is at most 2.

We will sketch the proof in the following steps:
• By Jorgensen and Thurston, there is a constant K>0 so that any hyperbolic 3-manifold M is obtained by Dehn filling a manifold X, so that X can be triangulated using at most K Vol(M) tetrahedra. We note that this is the only property of hyperbolic 3-manifolds we will use.
• Following Rieck and Sedgwick, if M is obtained from X as a generic Dehn filling, then every Heegaard surface F for M is a Heegaard surface for X. The term generic appearing in the statement of the theorem refers to this.
• Finally, we show that if F is a Heegaard surface for X of high genus then its distance is at most 2. It follows that F (as a Heegaard surface of M) has distance at most 2 as well. This follows from the following result, which is of independent interest and is a strong version of a result of Schleimer:

THEOREM: Let X be a manifold that admits a triangulation using at most t tetrahedra and F a Heegaard surface for X. If g(F) \geq 76t +26, then d(F) \leq 2.

We will conclude the talk with the idea of the proof.

Abstract : If a hyperbolic knot in a 3-manifold of Heegaard genus g admits a Dehn surgery to S^3, then how large can its genus g bridge number be In joint work with Cameron Gordon and John Luecke, we show that there are genus 2 manifolds that contain an infinite family of knots with integral surgery to S^3 such that the set of their genus 2 bridge numbers is unbounded. This contrasts our earlier work for non-longitudinal surgeries. In this talk we will focus on how to determine when the set of bridge numbers of a family of knots created by twisting along an annulus is unbounded. Then we will discuss its application to the result above.

Abstract:
We begin with a brief introduction to the Novikov homology and circle-valued Morse theory.
Then I will explain our joint work with Toshitake Kohno about the applications of the Novikov homology to the jump loci in the homology with local coefficients, in particular on compact Kaehler manifolds.

Abstract:
Thurston described a graph with a vertex for every closed oriented
3-manifold and an edge between two vertices $v_M$ and $v_N$ if there
is a Dehn surgery along a curve in $M$ yields $N.$ Famous results of
Lickorish and Wallace show this graph is connected and following those
constructions we may ask how few edges are needed to get from the
three sphere $S^3$ to a manifold $M$. After providing some background,
I will show that an infinite family of hyperbolic manifolds are not
path length one from $S^3$ even though they meet certain necessary
conditions like having cyclic homology. This is joint work with
Genevieve Walsh.

スライド（Slides）

アブストラクト：
1月の蒲谷さんのセミナーを受けて，Friedl-Vidussi による
ねじれアレキサンダー多項式を用いた3次元多様体のファイバー性の

ねじれアレキサンダー不変量の定義から始めます．

#### 2012年度

アブストラクト：There are many interesting and difficult algorithmic problems in low-dimensional topology. In the first part of this talk, we study the problem of finding a taut structure on a 3-manifold triangulation, whose existence has implications for both the geometry and combinatorics of the triangulation. We prove that detecting taut structures is “hard”, in the sense that it is NP-complete. We also prove that detecting taut structures is “not too hard”, by showing it to be fixed-parameter tractable. In the second part of the talk, we discuss ongoing efforts to extend these results to fundamental problems such as unknot recognition and prime decomposition of 3-manifolds. We base our work on Haken’s theory of normal surfaces, and we discuss both the theoretical and experimental complications that arise.
This is joint work with Joao Paixao and Jonathan Spreer.

Title : Virtual Fibering 予想について (I. Agol の仕事の解説)

Abstract:
“すべての双曲３次元多様体（完備で体積有限）は有限の被覆を取る事でS^1上の曲面束の構造を持つか？”という問題はvirtual fibering予想として知られている。
この予想はThurstonがBulletin of AMSの論文で提出した24の問題の中でも特に不思議な問題と思われる。
（Thurston自身”This dubious-sounding question …”と書いている。）
Thurstonは他にもvirtual Haken予想(Waldhausenの予想)，virtual b_1>0予想を提出しているが，その中でもvirtual fibering予想が最も強い形のもので，これらの関連する予想はvirtual fibering予想から従う。

さらに2012年にはAgolにより任意の閉双曲３次元多様体についてもvirtaul fibering予想が正しい事が示された。

とくに2008年のAgolの論文 “Criteria for virtual fibering” の解説を重点的に行う予定である。
この結果と2012年のAgolの定理 “cubulated hyperbolic groups are virtually special” からどのように閉双曲３次元多様体のvirtual fibering予想が従うかについて解説する。

#### ノート

Title : チェーン絡み目の例外的手術について (Martelli-Petronio, Martellli-Petronio-Roukema の仕事の解説)

Abstract:
Martelli-Petronio は，下記論文 [1]において，(3-)チェーン絡み目の例外的手術を完全に決定した．さらに，Martelli-Petronio-Roukema の 3 氏は [2] において，（minimally twisted) 5-チェーン絡み目の例外的手術を完全に決定した．これらの絡み目は，それぞれ 3成分，5 成分の双曲的絡み目の中で，双曲体積が最小であると予想されており，「重要」な双曲的多様体の多くを Dehn手術によって生み出すことが出来る．[1] の帰結は，概ね次のステップにより得られている．
Step 1: Gromov の 2π-theorem により，双曲的 Dehn 手術を与える「ほとんど」のスロープを決定する．
Step 2: Gromov の 2π-theorem の仮定からは漏れるが双曲的 Dehn 手術を与えるスロープを SnapPea を用いて決定する．
Step 3: Steps 1, 2 で除外できなかった スロープに沿った Dehn 手術で得られる多様体を，スパインを使って完全に決定する．
Step 3 では，スパインの変形を用いた ad hoc な議論が使われているが，著者の両氏は，彼らによる 3 次元多様体の “brick”への分解を念頭に置いて，確信を持って議論を展開しているように思われる．

[1] B. Martelli and C. Petronio, Dehn Filling of the “Magic” 3-manifold, Com m . Anal. Geom 14 (2006), 969–1026
[2] B. Martelli and C. Petronio, F. Roukema, Exceptional Dehn surgery on the minimally twisted five-chain link, arXiv:1109.0903

・井戸絢子（奈良女子大学大学院人間文化研究科）

Title :
Heegaard splitting of distance exactly n for each non-negative integer n
（joint work with Yeonhee Jang and Tsuyoshi Kobayashi）

Abstract:
Hempel introduced the concept of distance of Heegaard splitting by
using curve complex, and showed that there exist Heegaard splittings
of closed orientable 3-manifolds with distance $>n$ for any integer $n$.
In this talk, we construct pairs of curves with distance exactly
$n$ for any integer $n$, and we show that there exist Heegaard
splittings of 3-manifolds with distance exactly $n$.

・Michael Yoshizawa (University of California, Santa Barbara)

Title : Generating Examples of High Distance Heegaard Splittings

Abstract:
Given a closed orientable 3-manifold M, a surface S in M is a Heegaard
surface if it separates the manifold into two handlebodies of equal genus.
This decomposition is called a Heegaard splitting of M. The Hempel
distance of this splitting is the length of the shortest path in the curve
complex of S between the disk complexes of the two handlebodies. In 2004,
Evans developed an iterative process to construct a manifold that admits a
Heegaard splitting with arbitrarily high distance. We first provide an
introduction to Hempel distance and then improve on Evans’ results.

Andrei Pajitnov (Universite de Nantes)

Title : The Novikov homology and complex hyperplane arrangements

Abstract:
In the first part of the talk we give a brief survey
of the Novikov homology and its applications to
to the topology of circle-valued Morse functions
and more generally, closed 1-forms. In the second part
we compute the Novikov homology of the complement
to a complex hyperplane arrangement. We show that
for an essential arrangement in an n-dimensional
complex vector space the Novikov homology of its
complement vanishes in all degrees except n.
This is a joint work with Toshitake Kohno.

#### 2011年度

Title: Lectures on group orderings in low-dimensional Topology

Abstract:
Recently, the relationships between invariant group orderings and low-
dimensional topology receives much attention. It has been observed that
group orderings are related to Foliation (Lamination), Dynamical systems,
Rigidity theories, and Heegaard Floer theories and various other areas.
The subject “Ordering and Topology” are growing up rapidly, but at this
time there are no available textbook or survey. In this talk I will give
a short lecture on this wonderful new field of mathematics, and give an
overview of this subject:

* Basics of group orderings
* Ordering of 3-manifold groups
* Research Problems and recent progresses

Poster

Benjamin Burton（The University of Queensland）

Title: Knot invariants, normal surfaces and integer programming

Abstract:
The crosscap number of a knot is an invariant that is difficult to compute, and for which no general algorithm is known.
In this talk we discuss two new methods for computing crosscap numbers based on normal surfaces.
One involves a streamlined enumeration of the Hilbert basis for a pointed rational cone; the other formulates the problem using exact integer programming.
Although both methods may give indetermi- nate results in some cases, in practice they yield 191 new crosscap numbers that were previously unknown.

Poster

Vincent Blanloeil　（University of Strasbourg）

タイトル：”Pull back relation of knots”

タイトル：SnapPeaについて

アブストラクト：
SnapPea は J. Weeks によって作られたコンピュータプログラムで，
３次元多様体が四面体分割で与えられた場合にその双曲構造を求めてくれる。
また双曲構造が求まった場合には，体積や双曲デーン手術の計算に利用できる。

とくに双曲多様体の標準分解の理論を利用する事で，

SnapPeaのデータの構造について解説する。

タイトル：
「ねじれアレキサンダー多項式のこれまでと最近の話題について」

アブストラクト：
ねじれアレキサンダー多項式とは, 結び目群の線形表現を利用したアレキサンダー多項式の

スライス性の評価などに利用でき、より強力な判定条件を与えることが知られている。

「A Survey of Twisted Alexander Polynomials」
(S. Friedl and S. Vidussi, http://arxiv.org/abs/0905.0591 から, またはFriedl, Vidussi両氏のホームページから入手可能)

「Twisted Alexander Polynomials of hyperbolic knots」
(N. Dunfield, S. Friedl and N. Jackson,http://arxiv.org/abs/1108.3045 から入手可能 )

Teruhisa Kadokami (East China Normal University)

Title:
Hyperbolicity and identification of Berge knots of types VII and VIII

Abstract:
T. Saito and M. Teragaito asked in their paper whether Berge
knots of type VII, which can be situated on the fiber surface
of the lefthand trefoil, are hyperbolic, and showed that some
infinite sequences of the knots are all hyperbolic. We show
that Berge knots of types VII and VIII are hyperbolic except
the known sequence of torus knots. We used the Reidemeister
torsions. Consequently, the Alexander polynomials of them
have already shown their hyperbolicities. We also show that
the standard parameters identify Berge knots of types VII
and VIII up to orientations and mirror images, and study what
kind of information identify them.

Yo’av Rieck (University of Arkansas)

The Link Volume of 3-Manifolds II (joint woth with Yasushi Yamashita)

#### 2010年度

１３：３０〜１５：００
Marion Moore (University of California at Davis)
Title: High Distance Knots in closed 3-manifolds

Slides

Abstract:
Let M be a closed 3-manifold with a given Heegaard splitting. We show that after a single stabilization, some core of the stabilized splitting has arbitrarily high distance with respect to the splitting surface. This generalizes a result of Minsky, Moriah, and Schleimer for knots in S^3. We also show that in the complex of curves, handlebody sets are either coarsely distinct or identical. We define the coarse mapping class group of a Heeegaard splitting, and show that if (S, V, W) is a Heegaard splitting of genus greater than or equal to 2, then the coarse mapping class group of (S,V,W) is isomorphic to the mapping class group of (S, V, W). This is joint work with Matt Rathbun.

１５：３０〜１７：３０
Yo’av Rieck (University of Arkansas)
Title: On the tetrahedral number of hyperbolic manifolds of bounded volume
(after Jorgensen and Thurston, joint with Tsuyoshi Kobayashi)

Abstract:
In the 70’s, Jorgensen and Thruston proved that for any V>0 there exists a finite collection of manifolds X_1,…,X_n so that any complete hyperbolic 3-manifold of volume at most V is obtained by filling some X_i. A well-know “folk theorem” of Thruston says that there exists a constant K so that X_i can be triangulated using at most KV tetrahedra.
We will first motivate this theorem by describing two applications. The purpose of this talk is providing a proof of Thurston’s theorem. The proof follows an outline that appeared in the litrature, but as remarked by Benedetti and Petronio, it requirs control over the intersection between the Voronoi cells and the thin and thick parts of the manifolds (the terms will be explained in the talk). We will show how we control these intersections.
Most of the work is elementary and done in hyperbolic 3-space. I will make an effort to make it accessible to students familiar with the upper half space model.

9:00~13:00

9:00~11:00

Properties of Gauss phrase and category of regions
(桐生裕介氏 (Studio Phones) との共同研究)

11:00~13:00

Juxtaposing all the definable symbols

14：00~

On exceptional surgeries on Montesinos knots
(joint work with In Dae JONG (OCAMI) and
Shigeru Mizushima (Tokyo Institute of Technology))

Slides

＊上記講演終了後、自由講演を予定しています。

#### 2009年度

１５：００～１６：００

１６：１５～１７：１５

１０：３０～１２：００
Tunnel complexes of 3-manifolds

１３：００～１４：３０
Band surgery from (2, 6)-torus link to 7 crossing knot
(jont work with 下川航也)

１５：００～１６：３０
(joint woth with Yasushi Yamashita)
Yo’av Rieck (University of Arkansas)

#### 2008年度

プログラム

13:30–15:00 In Dae JONG (大阪市立大学)

(joint work with Kazuhiro Ichihara (Nara University of Education) and
Shigeru Mizushima (Tokyo Institute of Technology))

15:30–17:00 Andrei Pajitnov (Université de Nates)

10:30–12:30 Yo’av Rieck (University of Arkansas)
A linear bound on the genera of Heegaard splittings with distances
greater than two
(joint work with Tsuyoshi Kobayashi)

14:00–16:00 Joseph Maher (Oklahoma State University)
Random walks on the mapping class group

#### 2007年度

10:00～11:00　石原 海（埼玉大学）
Algorithm for determining depth and degree of tunnels

11:20～12:20　蒲谷 祐一（東京工業大学）
Finding ideal points from an ideal triangulation

14:00～15:00　中江 康晴（東京大学）
A good presentation of fundamental group of (-2,p,q)-pretzel knot and Reebless foliation

15:20～16:20　Arnaud Deruelle（東京工業大学）
Seiferters and Covering knots

16:40～17:40　Thomas Mattman（カリフォルニア州立大学）
Commensurability classes of (-2,3,n) pretzel knot complements

#### 2006年度

14:00–14:40
Makoto Ozawa　(Komazawa University)
Essential state surfaces for knots and links

15:00–15:40
Masaharu Ishikawa (Tokyo Institute of Technology)
Legendrian graphs and quasipositive diagrams

16:00–17:00
William W. Menasco (SUNY at Buffalo)
Legendrian knots and rectangular diagrams

#### 2005年度

アブストラクト：
After a recollection of the classical
Morse theory, in particular the construction of the
Morse complex, we shall discuss the Novikov complex –
the analog of the Morse complex for the more general
case of Morse maps to the circle. This complex is
closely related to the invariants of the gradient flow
of the Morse map, in particular it turns out
that the Lefschetz zeta function of the gradient
flow can be computed in terms of homotopy invariants
of the Novikov complex.

アブストラクト：
コンパクト多様体上のリーマン計量全体のつくる空間上で，
スカラー曲率の全積分で定義される汎関数（i.e.，Einstein-Hilbert 汎関数）を考え，
それを用いて山辺不変量と言う微分位相不変量が定義される．
この講演では，山辺不変量の基本的な結果から解説して，

#### 2004年度

プログラム：

１０時３０分ー

(-2,m,n)-pretzel knots
(門上晃久氏（大阪市大）との共同研究）

そして Seiberg-Witten 不変量の計算
(松田浩氏（広島大）との共同研究）

#### 2003年度

タイトル：Floer ホモロジー理論へのいざない

#### 2002年度

６号館２階６２４教室(午後)

１０：３０ー１２：００

Genera and boundary slopes of surfaces in 2-bridge link exterior (joint work with Hiroshi Goda (Tokyo A & T) and Hyun-Jong Song (Pukyong National Univ.))

１３：３０ー１４：３０

Novikov inequality for knots and links

１５：００ー１７：００

Exceptional surgery and boundary slopes (joint with Masaharu Ishikawa (Tokyo Metropolitan Univ.) and Thomas W. Mattman (California State Univ., Chico))

#### 2001年度

Double covers and Heegaard splitting (午後)

abstract: We will talk about handle decompositions of a
3-manifold (with boundary), coming from the Heegaard splittings
of the manifold, and describe a characterization,
which is read from the Heegaard splitting, for the corresponding
handle decomposition to be in thin position.