Tsuyoshi KOBAYASHI Home Page

2007 Nara Topology Seminar

11月20日（火）16:20-17:50

場所：奈良女子大学理学部新B棟４階　数学セミナー室３

講演者：井上 歩氏 (東京工業大学)

講演題目：quandle による結び目不変量の構成について
(On constructing knot invariants via quandles)

Abstract:
quandle とは集合とその上に定義された二項演算の組である条件を満たすものです．
この条件は群において積演算を忘れ共役の性質を残したものと考えることができます．
quandle の構造は様々な対象に対して導入することができますが，特に（任意次元の）
結び目に対しても定義することができます．
この結び目に対して定義された quandle は結び目群よりも強力な不変量であり，
古典的結び目に対しては完全不変量であることが知られています．また，quandle には
群と同様の手法でホモロジー／コホモロジーを定義することができ，そのサイクル／コ
サイクルを用いて古典的及び曲面結び目の不変量を定義することができます．
この講演では，結び目の不変量を構成することを目標にquandle についての紹介を行い
たいと思います．前提知識としては，少しだけ一般次元の結び目や曲面結び目にも触れ
ますが，基本的には古典的結び目の定義と Reidemeister moves を知っていれば十分だ
と思います．

Abstract:
A quandle is a pair of a set and whose binary operation which satisfies some
conditions. Where the conditions is considered as properties of conjugations
of a group forgetting properties of products. We could find quandle structures
for diverse mathematical objects.
In particular, for any dimensional knot, we could define the quandle of a knot.
It is known that this quandle is stronger than the knot group of the knot and
complete invariant for classical case. On the other hand, as a group, we could
define homology / cohomology groups of a quandle. Furthermore, we could define
invariants of a classical or surface knot with cycles / cocycles of quandles.
In this talk, I will introduce quandles and how to construct invariants of a
knot via quandles. It may be sufficient to understand my talk, if you know
definitions of a classical knot and the Reidemeister moves.

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10月29日（月）14:40-16:10

場所：奈良女子大学理学部新B棟４階　数学セミナー室３

講演者：金英子氏(東京工業大学)

講演題目：A property of the dilatation spectrum of the chain-link
with 3 components

Abstract: Let M be a hyperbolic 3-manifold which admits surface bundle structures
over the circle. An algebraic integer, called the dilatation, is associated
to each bundle structure of M. We consider the set of dilatations associated
to all bundle structures of M, called the dilatation spectrum of M. We show
that the dilatation spectrum of $S^3 \setminus C_3$, the complement of the
chain-link with 3 components $C_3$ in the 3-sphere, contains two subsequences
such that one converges to 2 and the other converges to 1.

We also show that $S^3 \setminus C^3$ admits an n-punctured disk bundle
structure over the circle for each integer n greater than 3. This tells us
that the minimal volume among all n-punctured disk bundles over the circle
with 3 cusps is bounded above by the volume of $S^3 \setminus C_3$.

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10月29日（月）16:20-17:50

場所：奈良女子大学理学部新B棟４階　数学セミナー室３

講演者：高沢光彦氏(東京工業大学)

講演題目：いくつかの計算機実験について
（上記の金氏の講演に関連したお話です．）

Abstract: 曲面束の hyperbolic volume と dilatation に関連するいくつかの計算機実験と、
そこから得られた観察を紹介します。特に、以下の２つに重点をおくつもりです。

無限個のファイバー構造を許容する曲面束におけるdilatation とファイバーのオイラー
数については一定の関係がある事が理論的にわかっています。３成分の chain link の
補空間 M において一部のファイバー構造に対して具体的な計算を行いました。

最小の dilatation を持つと予想されている組紐の族があり、そこから曲面束の族
を構成する事が出来ます。それら全てが、実は一つの多様体 M をデーン手術して得ら
れることがわかりました。

自分で計算機を使うのは敷居が高いなぁと思っている方にも親しんでもらえるように、
初歩的な計算機の利用から、複雑な実験までを、実例を用いて解説する予定です。

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10月26日（金）13:00-14:00, 16：30-17：30
 （時間が分かれていることにご注意ください．）

場所：奈良女子大学理学部新B棟１階　数学セミナー室５

講演者：茂手木公彦氏 (日本大学)

講演題目： "Networking Seifert surgeries on knots"
(joint work with Arnaud Deruelle and Katura Miyazaki)

Abstract: How can we obtain Seifert fibered surgeries on hyperbolic knots?
In this talk, we will propose a new approach to this question.
We introduce the "Seifert Surgery Network" consisting of all the
integral Dehn surgeries on knots in the 3-sphere yielding Seifert
fiber spaces, where Seifert fiber spaces may have fibers of indices
zero as a degenerate case.
We will start with some general results and then discuss the connectivity
of the network. In particular, we will discuss which surgeries on torus
knots can be "spreaders" in the Seifert Surgery Network.

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3月17日（土）15:00-16:00

場所：奈良女子大学理学部Ｃ棟４階　C431-2 （数学演習室）

講演者：Prof. Sung Sook Kim(Paichai University)

講演題目： The Nielsen Numbers and Minimal sets of Periods
for maps on the Klein bottle

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Date: March 17(Sat.) 15:00-16:00

Room: Room: Nara Women's University C434

Speaker: Prof. Sung Sook Kim(Paichai University)

Title: The Nielsen Numbers and Minimal sets of Periods
for maps on the Klein bottle

Abstract: In this talk, we concern with the self maps on the Klein bottle.
In 1911, Bieberbach proved that any automorphism of a
crystallographic group is conjugation by an element of Aff($\R)=\R
\rtimes \GL(n, \R)$. This was generalized to almost
crystallographic group. In 1995, K. B. Lee generalized this
result to all homomorphisms from isomorphisms. It can be state
as every endomorphism of flat manifolds is semi-conjugate to an
affine endomorphism. We can restate K. B. Lee's results in Klein
bottle group case as follows:

Let $\pi, \pi'\subset$ Aff($G$) be two Klein bottle groups. Then
for any homomorphism $\theta : \pi \to \pi'$, there exists
$g=(d,D) \in aff(G)= G \rx End(G)$ such that
$\theta(\alpha)\cdot g = g \cdot \alpha$ for all $\alpha \in \pi$.

Let $f: K \to K$ be any continous map on Klein bottle $K$ with the
holonomy group $\Z_2$ and let $\theta : \pi \to \pi$ be the induced
homomorphism on the fundamental group. We obtain two types of
$g=(d,D)$ by the semi-conjugate condition, and we calculate the
Nielsen numbers of periods for maps on the Klein bottle.

In terms of the Nielsen numbers of their iterates, we totally
determine the minimal sets of periods for all homotopy classes of
self maps on the Klein bottle.

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3月17日（土）16:20-17:20

場所：奈良女子大学理学部Ｃ棟４階　C431-2 （数学演習室）

講演者：小林 毅(奈良女子大学)

講演題目： Distances of knots and Morimoto's Conjecture on the super
additive phenomena of tunnel numbers of knots

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Date: March 17(Sat.) 16:20-17:20

Room: Nara Women's University C434

Speaker: Tsuyoshi Kobayashi(Nara Women's Univ.)

Title: Distances of knots and Morimoto's Conjecture on the super
additive phenomena of tunnel numbers of knots

Abstract: Let $K_i$ ($i=1,2$) be knots in the 3-sphere $S^3$, and
$K_1 \# K_2$ their connected sum. We use the notation $t(\cdot)$
to denote tunnel number of a knot. It is well known that the
following inequality holds in general.

$$t(K_1 \# K_2) \leq t(K_1) + t(K_2) +1.$$

We say that a knot $K$ in a closed orientable manifold $M$ admits a
$(g,n)$ position if there exists a genus $g$ Heegaard surface
$\sigma \subset M$, separating $M$ into the handlebodies
$H_1$ and $H_2$, so that $H_i \cap K$ ($i=1,2$) consists of $n$
arcs that are simultaneously parallel into $\partial H_i$. It is
known that if $K_i$ ($i=1$ or 2) admits a $(t(K_i),1)$ position
then equality does not hold in the above. Morimoto proved that if
$K_1$ and $K_2$ are m-small knots then the converse holds, and
conjectured that this is true in general (K.Morimoto, Math. Ann.,
317(3):489--508, 2000).

Morimoto's Conjecture
Given knots $K_1,\ K_2 \subset S^3$, $t(E(K_1 \# K_2)) < t(E(K_1)) +
t(E(K_2))+1$ if and only if for $i=1$ or $i=2$, $K_i$ admits a
$(t(K_i),1)$ position.

In this talk, we describe how to show the existence of conterexamples
to this conjecture by making use of the \lq distance\rq of knots.

2月21日（水）16:30-17:30

場所：奈良女子大学理学部Ｃ棟４階　C431 （数学演習室）

講演者：下川航也氏(埼玉大学)

講演題目：DNA and lens space surgery

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Date: February 21 (Wed.) 16:30-17:30

Room: Nara Women's University C431

Speaker: Prof. Koya Shimokawa(Saitama University)

Title: DNA and lens space surgery

Abstract: In this survey talk I will show how lens space surgeries on knots
in the 3-sphere can be applied to the study of enzymes acting on DNA.
In particular, I will discuss how our result (joint with Hirasawa)
on lens space surgeries on strongly invertible knots has been put
to use by biologists.

大阪大学・奈良女子大学合同トポロジーセミナー

1月29日（月）15:30-17:00

場所：奈良女子大学理学部Ｃ棟４階　C431 （数学演習室）

講演者：Prof. Jonathan Hillman(シドニー大学)

講演題目：Finiteness conditions, Novikov rings and Mapping Tori

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Date: January 29 (Mon) 15:30-17:00

Room: Nara Women's University C431

Speaker: Prof. Jonathan Hillman(University of Sydney)

Title: Finiteness conditions, Novikov rings and Mapping Tori

Abstract: In 1962 Stallings gave a simple and natural algebraiccriterion for
a 3-manifold to be a mapping torus, in other words, to fibre over
the circle. The corresponding infinite cyclic covering space is then
homotopy equivalent to a closed surface, and so is a $PD_2$-complex.
An analogous result for infinite cyclic coverings of higher dimension
manifolds with fundamental group $Z$ was found by Milnor, and Quinn
and Gottlieb extended this to more general fibrations of $PD$-complexes
over $PD$-complexes.
We are interested in simplifying the hypotheses of such homotopy
fibration theorems. Our main result is the following:
Theorem. Let $p:M'\to M$ be an infinite cyclic coverof a closed
$n$-manifold (with $n\not=4$). Then $M'$ is a $PD_{n-1}$-complex if
and only if $\chi(M)=0$ and $M'$ has finite $[(n-1)/2]$-skeleton (up to homotopy).
The theorem remains true for $n=4$ if we replace ``$PD_3$-complex" by
a slightly weaker notion.
This result may be regarded as lying between those of Milnor and
Gottlieb-Quinn. Our arguments are purely homological, and take full
advantage of Poincare duality in $M$. Examples deriving from high
dimensional simple knots show that the dimension bounds are best
possible.
This is joint work with D.H.Kochloukova, of Brazil. I have not met her;
our collaboration is a happy consequence of the travels of Peter Zvengrowski.

大阪大学・奈良女子大学合同トポロジーセミナー

1月8日（月）15:30-17:00（この日は祝日です）

場所：奈良女子大学理学部Ｃ棟４階　C431 （数学演習室）

講演者：Prof. Craig Hodgson(メルボルン大学)

講演題目：Introduction to hyperbolic 3-manifolds

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Date: January 8 (Mon) 15:30-17:00

Room: Nara Women's University C431

Speaker: Prof. Craig Hodgson(The University of Melbourne)

Title: Introduction to hyperbolic 3-manifolds

Abstract: This talk will give an introduction to hyperbolic geometry, describe
examples of hyperbolic 3-manifolds, and survey some of the main
results on the classification of hyperbolic 3-manifolds. The talk is
aimed at a general audience, and should be suitable for beginning
graduate students.